By car

The main road through the park is open to automobile traffic only as far as Savage River (mile 14). Travel beyond this point is allowed only on the park shuttle buses, on foot, or on a bicycle.
For people camping at Teklanika campground, one automobile per a campsite is allowed to drive in to it.
For those visiting Denali in mid-September the park service opens the road completely for four days after the shuttle buses stop running. Only 400 vehicles per day are admitted, and each vehicle requires a special permit. To potentially obtain one of these permits contact the park and inquire about the Denali Road Lottery. The park service will ask you to mail your address, preference of dates, and a fee during the month of July. In mid-August you will be contacted if you have been selected. If snow has not yet closed the road, once the four-day lottery is complete the road will be opened to private vehicles for travel as far as Teklanika Campground (mile 29) until weather closes it for the season. It is also important to note that the road may close during the road lottery if inclement weather shows up - much to the chagrin of all those people who won a permit only to have it canceled.

By shuttle bus

Shuttle buses are allowed past Savage River (mile 15) on the park road, as far as Kantishna (the end of the road). Passengers may disembark from the buses at any point west of mile 20, and then re-board any bus on a space-available basis. Anyone disembarking should be aware of the bus schedule, and plan for at least a one hour wait for a bus with seats available to arrive.
A very common question people have about the buses is, "So what's the difference between a shuttle bus and a tour bus." Simply put, most of the people on tour buses are on packaged trips with the local hotels, and they get a dedicated naturalist on their bus that is required to give commentary throughout the tour. Buying tickets for tours can be more difficult because the vast majority of the tickets are reserved when people book their packaged vacations. However, they are not necessarily "better" than a shuttle bus. Two important facts to note are that all the buses drive the same road (duh, there's only one road inside the park!), and that all the buses have similar destinations. By similar destinations I mean that there's a tour bus to Eilson Visitor Center and there is also a shuttle bus to Eilson Visitor Center. So, if you ask, "Which bus is better for seeing wildlife?" The answer is that they are about the same since they are on the same road going to the same place. Your experience on the shuttle bus happens to vary quite a bit depending on the driver you get. Some of the shuttle bus drivers will talk just as much as a tour bus driver, however, they are not required to. Some of the shuttle bus drivers won't say much of anything unless somebody asks them a question. It's important to note that a pretty large majority of the bus drivers will give some form of commentary as they drive, because they want to share their love of the park just as much as all the other employees. Oftentimes, a deciding factor for people on whether to take a shuttle or a tour to Eilson is, "Do I want to pay about $30 for a shuttle bus or about $95 for a tour bus?" You be the judge.
Shuttle bus reservations can be made either by following the instructions at the park concessionaire web site or in person at the reservation desk in the visitor center. Be aware that buses may fill several days in advance, especially during the height of the summer visitor season.

By foot

There are few trails within the park, but visitors are allowed (and often encouraged) to choose their own paths across the tundra. The less-adventurous may choose to simply amble along the park road after Savage River; keep an eye out for buses and wildlife when traveling on the road. Be aware that Denali hosts a large, robust population of brown bears and that they can be anywhere and sow bears are extremely fierce when they feel their cubs are threatened in any way. Always travel in groups and don't go into the backcountry without someone, preferably the park itself, knowing your plans. No one will come looking for you if they don't know you are lost, and Denali is not a place you want to be lost and alone.

By bike

A mountain bike is a great option for traveling on the park road. Sometimes bikers arrange backcountry permits at the Backcountry Information Center that allow them to spend a few days traveling out to Wonder Lake and back. This can, however, be logistically tricky - you will need to either spend the night in an established campground or be near enough to one that you can stash your bike (and food if you are not carrying a Bear Resistant Food Canister) overnight. The only areas where this is possible are: Wonder Lake; Toklat River (you can't sleep there, but you can use the food lockers at the temporary ranger contact station); Igloo Campground (again, you can't stay there, but you can utilize the food lockers if need be); Teklanika Campground; and Savage Campground.
If planning a trip by bike along the park road be prepared for travel on a dirt road with several major mountain passes and few guard rails.
One fun option is to take a bike out to Wonder Lake on a camper bus. You need to tell the person who you buy the ticket from that you plan to take a bike since only two bikes are allowed per bus. Once you get to Wonder Lake, you bike back out of the park. This trip can be done in approximately 10 hours if you keep a good pace. It is especially enjoyable if you plan this bike trip around the same time as summer solstice. You can take the last bus going out to Wonder Lake so that you can bike the entire trip back with no buses on the road, all while getting to experience an awe-inspiring bike ride in the land of the midnight sun.