The university operates several free shuttle lines, including lines from Lawrence to the academic portions of campus, other graduate housing complexes, and even some local shopping centers. For more information, consult the Tiger Transit website.
Walking to the Dinky Station on campus takes between 10 and 20 minutes for most people. Walking to the academic portion of campus takes between 20 and 45 minutes, depending on where you are going. Be sure to study a map first--cutting diagonally across campus can save a lot of time and is much prettier than walking along the major streets!
Many people in Lawrence have a bike. There are several cycling stores in town where you can get a decent bike for around $300. (Most services in Princeton are expensive, and bike repairs are no exception. Consider this before getting a flimsy bike from Wal-Mart for $100.) You can also purchase used bikes on TigerTrade. Every fall and spring, the university has a bike giveaway. The bikes are those found abandoned on the racks around campus, so they are of varying quality--if you want a decent bike, make sure to get there early!
Whether you should buy a car depends on your habits and budget. Single students with little grocery shopping to do usually manage to get by without a car (with occasional help from others, e.g. when transporting furniture). If you are from outside the US, make sure that you want to pay the cost of insurance and repairs before buying a car. If you have a driver's license, you can also rent a car for occasional trips.