Many summer visitors visit Jackson Hole and Grand Teton National Park as a part of a larger road trip, often including Yellowstone and Glacier National Parks and even the Black Hills in South Dakota.
Here are a few distances to major cities to give you an idea of location:
Flying in is certainly the most convenient, especially in the winter months, but can add extra costs depending on how many are traveling in your party. Also keep in mind that a rental car is almost mandatory in the area during the summer, while you can get by on shuttle and public transportation in the winter.
There are a couple of companies that offer scheduled or private shuttles to/from Jackson Hole. Try Alltrans for shuttles to/from the Jackson Hole Airport and Salt Lake City Express for transportation to/from the Salt Lake City Airport.
While public transportation improves every year in Jackson, having your own set of wheels out here (espeically in the summer months) will offer more convenience on your vacation. Rental cars are available at all of the airports mentioned above. No 4wd needed in the summer but be prepared for winding mountain passes.
A rental RV is a great option if you are visiting in the summer and planning on camping the entire time and moving from site to site. The only downfall of a rental RV is that you have to pack it up every time you want to go somewhere. Camper/travel trailers make much more sense so that you can park it and use your more fuel efficient car to see the sites but unfortunately, most rental car companies will not allow you to tow anything. The most convenient locations to find rental RVs are Jackson and Bozeman.
If you are flying to Jackson Hole, only plan on seeing Jackson and Grand Teton National Park, have a flexible schedule, or are visiting in the winter, public transportation is a great option. The START bus in Jackson is great for getting around town (free) and out to Teton Village ($3 one way). It runs from early morning until about 10:30pm.
In the summer, Altrans offers a Grand Teton Park shuttle that leaves right from town and has many stops within the park. The earliest pickup starts at 7:30 and it runs a handful of times until about 7pm. $14 to ride (does not include $12 park entrance). In the winter, Altrans features a Grand Targhee Ski Area shuttle/lift ticket combo.
Flying to Jackson Hole Airport (JAC) is an easy way to access Jackson Hole. The airport is the only airport located in a national park (Grand Teton National Park) and thus offers tremendous views when you fly in.
Many also choose to fly in to Salt Lake City International Airport (SLC) and make the 4 hour 40 minute drive to Jackson over scenic mountain passes.
Driving, you can access Jackson Hole from any direction, but be aware that roads through Yellowstone National Park are closed from early November to varying times in May.