There are pros and cons to the various methods of getting around San Juan.

Rental car
Getting around various parts of the city, and the rest of the island, as well as to and from the airport, will be much more convenient with a car. Though road signs are in Spanish, the road sign shapes are going to be as familiar to you if you are used to American road signs. And when you see the word norte sitting above a numbered-road sign enough times, it won't take you long to realize that means north. However, a car may entail paying parking fees as you travel around to your destinations so you will need to budget for that. If you go to Old San Juan, parking may be expensive and quite difficult (see that article for how to find the few parking garages there).
Driving in San Juan is very similar to driving in Florida or the cities of the northeast U.S. like Boston or New York. People drive quickly, change lanes with little notice, and frequently tailgate, despite narrow streets. Natives of areas with more placid driving styles like the U.S. Midwest may find it frightening, though.
None of the rental car agencies are within walking distance of the terminals, and the shuttle pickup curbs are not clearly marked or signed, so don't be afraid to ask for help in finding them. Avis and Hertz have their lots located on the airport grounds less than a quarter of a mile from the terminals, and the shuttle trip to and from their facilities is quick and predictable. All other agencies, including Thrifty and Dollar, are located at least one mile away or farther out. You must budget that extra 20 minutes into your planning or you may miss your return flight.
If you are a resident of the United States, check with your auto insurer to see if it already covers you in PR - most do. However, while that means you can decline the collision insurance, you should still take the loss damage waiver (even though it's expensive) as auto collisions in Puerto Rico are common (you will see a lot of dented fenders) and parking spaces in PR are not as large or forgiving as those on the mainland.
Puerto Rico is still using simple paint to mark lanes rather than modern thermoplastic striping. Unfortunately, paint fades fast in the tropical heat and rain, so road markings are hard to see or completely worn away on many roads. In poorer neighborhoods of San Juan, look out for missing manhole covers and huge potholes.
Like much of Latin American and the Caribbean, proper street signs aren't on every corner of San Juan, which makes a good street map (with landmarks) or a GPS navigation system essential. Only

Condado

and Ocean Park have excellent street signage comparable to those taken for granted in mainland U.S. cities.
The closest place to refuel a rental car before returning it is the Puma gas station on the frontage road ("Calle Marginal") off eastbound PR-26 (Expreso Loiza), at the intersection with Calle Heriberto. Once you're done there, you can continue down the road to Thrifty or Dollar, or keep going to the next underpass to loop around and head west towards the airport.

By bus
The public bus system in San Juan is inexpensive. Only coins are accepted on board so stock up on those quarters. The bus stops are marked "PARADA", and the system is currently being reworked to fit in with a new train line. If you're planning on staying in and around a general area of San Juan, you may be able to get by on public transportation. When you see your bus coming, be sure to wave/flag it down otherwise it may just pass you by!
Take notice; contrary to some bus schedules (even ones posted at the bus stops themselves) that state buses may arrive every 15 minutes or so, service can be infrequent and very unpredictable. You may wait anywhere from 5 to 30 minutes for a bus to arrive. Rush hour buses can be very full. Additionally, traffic gets very heavy heading into Old San Juan, so once you board the bus you still are not guaranteed a quick trip. If going into Old San Juan, some useful bus routes include T3 (from Sagrado Coazón train station), T5 (from Isla Verde via Miramar), T21 (from Condado), and D53 (from Luis Muñoz Marín International Airport via Isla Verde and Condado). Old San Juan, near the harbor front, has a major bus station for catching numerous routes.
Visitors may find that bus routes directly to and from places of interest around San Juan do not exist, and that a transfer with additional waiting times are involved. Plan your trips wisely as most routes work Monday through Friday from 5:00 AM to 9:00 PM; and Saturdays and holidays from 6:00 AM to 8:00 PM. T3 and E40 express route are the only two routes that offer service on Sundays. The T3 route works every day from 5:00 AM to 11:00 PM; and the E40 from 5:00 AM to 8:00 PM. Only If you have little time to see/do what you'd like, you'd be better served using a taxi or renting a car.
Also, have in mind, if you are planning to get back to the airport by bus, some bus drivers may not let you board the bus if you carry-on more than one baggage. You may want to plan in advance to take a taxi back to the airport if your hotel has no shuttle. Carrying baggage on the city bus tends to be less of an issue when you initially head from the airport. Three routes serve Luis Muñoz Marín International Airport: E40 express route (to Piñero train station), T5, and D53 (to Old San Juan, Condado, Isla Verde, and Puerto Rico Convention Center).
Check with AMA to inquire about routes and times. Also, for train and buses time tables you can download Tren Urbano App (unofficial app) for iOS and Android phones.
There are a free trolley bus routes around Old San Juan - look out for the numbered signs.
As for getting around the rest of the island of Puerto Rico, there are no convenient buses connecting San Juan to the rest of the island. There are vans that are like group taxis with specific routes called "guaguas." You can inquire about trips outside of San Juan at the guagua terminal in Rio Piedras. But keep in mind, guaguas may make multiple stops to maximize revenue, and it could take ages to get you to where you are going on the island.

By ferry
The Cataño Ferry (La Lancha de Cataño) is a public ferry serving Old San Juan and Cataño. It crosses San Juan Bay every 15-30 minutes.

By train
San Juan is served by a rapid transit rail line called "Tren Urbano" (Urban Train). The line connects San Juan to the towns of Guaynabo and Bayamón, but it avoids points of interest like Old San Juan, Condado, and Isla Verde. But if you want to give it a spin anyway:
- Take a city bus to a station, e.g., T3 or the E10 express route from the Old San Juan Covadonga bus terminal to Sagrado Corazón train station.
- Explore the town of Río Piedras by getting off at Universidad or Río Piedras stations and making your way down the colorful Avenida Juan Ponce de León. Explore the side streets and alleys to discover some wonderful street art.
- Get to the airport by taking the E40 express route at Piñero station.
There are ticket machines accepting cash and credit cards at all stations. The magnetic tickets they issue are rechargeable and also usable to pay your fare on the bus. There are discounted fares for students, children and elderly people on both the buses and the train.
For train and buses time tables you can download Tren Urbano App (unofficial app) for iOS and Android phones.

By taxi
Taxis can be found hanging around hotels and the east end of Calle de la Fortaleza in Old San Juan. In theory, they are supposed to be metered (the rates are posted on doors), except for a selection of common tourist routes with fixed fares. In practice, cabbies are well aware that tourists tend to have no idea what those fixed fares are and charge according to what they feel like.
For any issues with cabs, contact the Tourist Transportation Department at 787-999-2100 ext. 4502 between 7:30 AM to 4:30 PM Monday through Friday.