By train and subway
Tokyo has one of the most extensive mass transit systems in the world. It is clean, safe and efficient - and confusing. The confusion arises from the fact that several distinct railway systems operate within Tokyo - the JR East network, the two subway networks, and various private lines - and different route maps show different systems. Avoid rush hours if possible; trains get overcrowded very easily.
The defining rail line in Tokyo is the JR Yamanote Line (山手線), which runs in a loop around central Tokyo; being inside the Yamanote loop is synonymous with being in the core of Tokyo. Almost all inter-regional JR lines and private lines start at a station on the Yamanote. JR's lines are color-coded, and the Yamanote is green. The JR Chuo (中央線, orange) and Sobu (総武線, yellow) lines run side-by-side, bisecting the Yamanote loop from Shinjuku on the west to Tokyo on the east. JR's other commuter lines, the Saikyo and Keihin-Tohoku, run off the rim of the Yamanote loop to the north and south. JR East has a good English information line, 050-2016-1603 or 03-3423-0111.
Tokyo has an extensive subway network with frequent trains, and these are primarily useful for getting around within the Yamanote loop. The Tokyo Metro runs nine lines: Ginza, Marunouchi, Hibiya, Tozai, Chiyoda, Yurakucho, Hanzomon, Namboku and Fukutoshin lines. Toei operates the Asakusa, Mita, Shinjuku, and Oedo lines. While the JR Yamanote Line is not a subway line, due to its importance as a major transportation artery in downtown Tokyo, it is usually featured on subway maps. In addition, there is a largely underground Rinkai Line, a private line which is operated by Tokyo Waterfront Area Rapid Transit or TWR, that passes through the island of Odaiba.
Announcements and signs are usually bilingual in Japanese and English, though in some areas frequented by tourists, signs in Korean and Chinese can also be seen.
A number of private commuter lines radiate from the Yamanote loop out into the outlying wards and suburbs, and almost all connect through directly to subway lines within the loop. The private lines are useful for day trips outside the city and are slightly cheaper than JR. Among these, the most important to visitors is arguably the Yurikamome which offers great views on the way to the island of Odaiba.
If you are using a smartphone you can benefit from a free, official Tokyo Subway Navigation application available in the app stores. It works offline and has a multilanguage interface, including English. It is very useful for quick route searches, but its lookup seems to be time-optimized and not cost-optimized. Also, it only covers the subway and doesn't cover, for example, Monorail. 

Fares and hours
Most tickets and passes are sold from automated vending machines. Almost all vending machines now have an "English" mode. Keep in mind that JR trains are free with a Japan Rail Pass.
Prepaid fare cards are convenient and highly recommended because they allow you to ride trains without having to read the sometimes Japanese-only fare maps to determine your fare. There are two brands of prepaid fare cards, JR East's Suica, and PASMO, offered by private (non-JR) lines. Functionally they are completely interchangeable and can be used on just about every subway, train and bus line in Tokyo (with the noted exception of JR's Shinkansen and limited express trains).
The fare cards are rechargeable "smart cards": you simply tap your card on the touchpad next to the turnstile as you go in, and do the same when going through to exit.
The older Passnet cards are not accepted anymore. If you still own some of these, you can exchange them for a PASMO or Suica card.
There are also some special tickets that allow unlimited travel, but most are unlikely to be useful to tourists unless you're planning to spend half your day on the train.

• The Tokunai Pass (都区内パス) is a one-day pass good for travel on JR lines anywhere in the 23 wards of Tokyo (including the entire Yamanote Line and many stations surrounding it). It is economical if you plan to make five or more train hops in one day. A variant is the Tokunai Free Kippu (都区内フリーきっぷ), which also includes a round-trip into Tokyo from stations in the surrounding prefectures. The Monorail And Tokunai Free Kippu is good for two days and includes a round-trip from Haneda Airport to central Tokyo.
• The Tokyo Free Kippu (東京フリーきっぷ) covers all JR, subway, and city bus lines within the 23 wards. It covers a number of areas that are not served by JR, such as Roppongi and Odaiba.
• The Holiday Pass (ホリデーパス) covers the entire JR network in the Tokyo metropolitan area, including Chiba, Kanagawa, Saitama, and west Tokyo. It is only available on weekends, national holidays and during summer vacation (July 20 through August 31).
• The Tokyo Subway Ticket provides unlimited travel on consecutive days on all Tokyo Metro lines and Toei Subway lines, which serve most of central Tokyo. It is available as a 1-day, 2-day, or 3-day ticket. It is only available to overseas visitors with foreign passports, and can only be purchased at specific places such as Narita, Haneda, and Yonago airports, and BIC CAMERA stores. It does not cover JR lines, including the Yamanote line.
• The Tokyo Metro 1-Day Open Ticket covers the Tokyo metro network with the exception of Toei and Ginza lines. Can be purchased in advance or from vending machines.

If you're paying a la carte, subway and train fares are based on distance. As a general rule of thumb, Tokyo Metro lines are cheapest, Toei lines are most expensive, and JR lines fall somewhere in the middle (but are usually cheaper than Metro for short trips, i.e. no more than 4 stations). Many of the private lines interoperate with the subways, which can occasionally make a single ride seem unreasonably expensive as you are in essence transferring to another line and fare system, even though you're still on the same train. E.g. changing between Metro subway line and Tokyu private line amounts to paying the sum of each fare. In addition, several patterns of transfer are listed as "Transfer Discount". When using Suica or PASMO, you can get all the transfer discounts automatically. At some transfer stations, you may need to pass through a special transfer gate (both for paper tickets and PASMO/Suica) which is colored orange - passing through the regular blue gate will not get you your transfer discount and if you have a paper ticket, you won't get it back. At some transfer points (e.g. Asakusa station) you may actually need to transfer on street level as the two stations (Metro Ginza Line and Toei Asakusa Line) are not physically connected and are about one block apart.
It pays to check your route beforehand. The Tokyo Transfer Guide by the Tokyo Metro and Toei subway companies is an online service that allows you to plan subway and train travel from point A to point B, based on time, cost, and transfers. This guide provides information for Tokyo only, and there are other sites which additionally cover the whole country. Some major stations have terminals providing information similar to the Tokyo Transfer Guide.
If you can't figure out how much it is to the destination, you can buy the cheapest ticket and pay the difference at the Fare Adjustment Machine (norikoshi) at the end. Most vending machines will let you buy a single ticket that covers a transfer between JR, subway and private lines, all the way to your destination, but working out how to do this may be a challenge if you are not familiar with the system. When transferring between systems, whether paying with tickets or smart cards, use the orange transfer gates to exit. Otherwise, you'll be charged full fare for both separate parts of your trip, instead of the cheaper transfer fare.
Most train lines in Tokyo run from around 05:00 to 01:00. During peak hours they run about once every three minutes; even during off-peak hours, it's less than ten minutes between trains. The only night when regular passenger services run overnight is for the New Year's Holiday on select lines.
For additional information for train travel in Japan generally, refer to the By rail section in the Japan article. 

By taxi
Taxis are very pricey but may be a value for groups of three or more. Also, if you miss your last train, you may not have another choice.
These examples are based on standard routing and traffic conditions, so your actual fare may vary in relation to the estimated fare.
Do not count on your taxi driver speaking English--or knowing more than the best-known locations, though most taxis have GPS "car navi" systems installed. The best and easiest thing to do is to prepare a map marked with where you want to go and point it out on the map to the taxi driver. If you are staying at a hotel, they will provide a map. If possible, get a business card, or print out the address in Japanese of any specific places you wish to go. However, because in Japan streets are often unmarked, if the taxi driver does not have GPS he may not be able to do more than take you to the general vicinity of where you want to go. Also, note that taxis can get caught in traffic jams. Tips are entirely optional.
Taxi rear left passenger doors are operated by the driver and open and close automatically. Don't open or close them yourself.

By car
Tokyo is a gigantic warren of narrow streets with no names, with sometimes slow-moving traffic and relatively limited and expensive parking. In this city with such an excellent mass transit system, you would need a good reason to want to drive around instead, unless you already have some familiarity with the city and its surroundings. While renting a car does make sense in Japan in some contexts (e.g. traveling between cities, visiting smaller towns or a rural onsen resort), in general, it is neither convenient nor economical to rent a car to get around metro Tokyo. Taxis are much more convenient if your budget allows it; walking or public transportation is much less expensive and given the difficulties of navigation and finding parking in popular areas, probably easier too.
If you do decide to plunge in and drive around by car, the main expressway serving Tokyo is the Shuto Expressway, abbreviated to Shutoko (首都高). The C1 Loop Line forms a circle around central Tokyo, similar in fashion to how the Yamanote Line does it by rail. When driving around, both on Shuto and on the regular streets, try to avoid the rush hour (with traffic moving into the center in the morning, and out towards suburbs in the evening). Check with the car rental agency whether English-language navigation system is available, as it will make a huge difference, and you will be able to use your car to outperform the public transportation on many occasions.
Driving on the Tokyo Expressway at night can be a pleasant and beautiful experience as you whiz through and around the Tokyo nightlife. When driving at night you should exercise caution and obey speed limits: Street racing over the Shutoko at night became popular in the '80s and '90s and still happens today, albeit on a less frequent basis. Street racers often concentrate their driving on the C1 Loop Line and the Bayshore (more popularly known as the Wangan) Line. "Competitors" sometimes hang out at parking and service areas on the Shutoko, especially the large Daikoku Parking Area at the intersection of the Bayshore Line and the K5 Daikoku Line in Yokohama.

By bus
The few areas within Tokyo that aren't easily accessible by train are served by various bus companies. Buses operating within 23 wards of Tokyo have a fixed fare regardless of distance, which is paid upon boarding from the front door. The fares are not transferable; however, most buses do accept Suica or PASMO fare cards. If you use a "Suica" or "PASMO" card to board a Toei Bus, you will receive a discount on your next Toei Bus ride as long as it is within 90 minutes of the previous ride. The fare can also be paid in hard currency of practically any denomination, which can also be used to charge your electronic fare card if you tell the driver you wish to do so. Compared to the trains, the buses run much less frequently, carry fewer passengers, and are much slower. This makes them amenable to the elderly residents of Tokyo, but rather inconvenient for travelers, who will also have to deal with lack of information in English and sometimes very well hidden bus stops.
The bus routes are named by a kanji based on a destination they serve and a number, e.g. "渋64" for the route between Shibuya (渋谷) and Nakano. The routes can be fairly complicated and are often not listed in detail at the bus stops; signs on the buses themselves often list only two or three main stops in addition to the origin and destination. Finding a bus going towards a main terminal like Shibuya or Shinjuku is therefore pretty easy, the other way around may be more difficult. You may find several different stops by different companies within a few meters, all sharing the same name; sometimes those are full-fledged waiting areas and sometimes only a signpost at a street corner. Inside the bus, the next stop is usually announced several times, sometimes by a taped voice and sometimes by a mumbling driver; recently taped announcements in English are used on some lines, but are still rare. In addition, the stop is displayed on an electronic display, but rarely in Latin writing. Some stops may have similar names, you should make sure you know the exact name of the stop you want to get off at. Drivers are usually very helpful, though aren't typically used to tourists and may be rather busy dealing with fares, passengers and traffic. Nevertheless, north-south routes are useful in the western side of the city since train lines (Odakyu, Keio, Chuo, and Seibu) tend to run east-west. Google Maps has begun adding bus transit information and is possibly the easiest way of planning a trip by bus. 

By ferry
The Tokyo Cruise Ship Company operates a series of Water Bus ferries along the Sumida River and in Tokyo Bay, connecting Asakusa, Hinode, Harumi, and Odaiba. The ferries feature a recorded tour announced in English, as well as Japanese and a trip on one, makes for a relaxing, leisurely way to see the waterfront areas of Tokyo. Of particular note is the super-futuristic Himiko ferry designed by anime and manga creator Leiji Matsumoto, which runs on the Asakusa-Odaiba Direct Line. You might want to arrive well before the departure time just in case tickets on the Himiko sell out!

By bicycle
Bicycles are very commonly used for local transport, but amenities like bicycle lanes are rare, drivers pay little heed to bikes and traffic can be very heavy on weekdays, so if you use a bicycle, do not be afraid to cycle on the sidewalk (everyone does). Keep in mind, however, that parts of Tokyo are surprisingly hilly, and it's a sweaty job pedaling around in the summer heat. Central Tokyo can still be covered fairly comfortably by bike on the weekends. Tokyo Great Cycling Tour offers a one day guided tour for biking around major tourist spots in Tokyo, like Marunouchi, Nihonbashi, Tsukiji, Odaiba, Tokyo tower, Imperial palace and so on.
Renting a bike is possible from some youth hostels, particularly around Asakusa, although it's not common. However, buying a simple single-speed roadster is fairly cheap, and comes complete with a built-in bicycle wheel lock system (this is what most Tokyoites use). An imported multiple-geared bike will be much more expensive so get a good lock, as bike theft is a common threat, although the problem is nowhere near as serious as in other countries. 

By foot
In this large city with such an efficient public transportation system, walking to get from point A to point B would seem a bit stupid at first glance. However, as the city is extremely safe even at night, walking in Tokyo can be a very pleasant experience. In some areas, walking can be much shorter than taking the subway and walking the transit (the whole Akasaka/Nagatacho/Roppongi area in the center is for instance very easily covered on foot). If you have the time, Shinjuku to Shibuya via Omotesando takes roughly one hour, Tokyo Station to Shinjuku would be a half a day walk, and the whole Yamanote line Grand Tour takes a long day.