Alice Springs is flat (except for Anzac Hill), so hire a bicycle.....its the best way to get around.  There are heaps of sealed paths and it doesn't waste time walking between places.  Also means you can easily visit attractions that are 3 or 4 km out of town.  And there is a sealed path all the way to Simpsons Gap (about 17km) if you're really enthusiastic.  Just make sure you take lots of water.  You can also walk around Alice Springs with no problem, but hiring a bicycle does make it faster to get around, and its fun!  You don't need to hire a car unless you plan on exploring outside of the town (like Ormiston Gorge, Ellery Creek waterhole, Uluru, etc).

You can hire a bike from a lot of the hotels and hostels in town.  You're not subjected to anyone else's schedule but your own and it's fairly cheap - about $30 for a full day. Longhorn Bikes is a local company operating from most hotels. They hire out mountain bikes and urban bikes and even tandem bikes. Riding to Simpsons Gap and/or the Desert Park or the Alice Springs Telegraph Station is delightful and there are places to lock up your bike at each. Bike riding is a fabulous way of getting around Alice Springs. 

There is pretty limited public transportation in the form of public bus service, you can get a schedule at the tourist office but buses only run about every 1-2 hours.  The bus departs from the Yeperenye Shopping Centre on Hartley Street. There is also the Alice Wanderer hop-on/hop-off bus tour which takes you to the major sights and gives you a running commentary on what you're seeing in town. This costs about $42 and the tour lasts about 70 minutes.