OP- The free shuttle is for Resort guests only, and goes to the airport and around the lodging/campground/shopping loop. There is a paid shuttle to the Cultural Center which is a few km from Uluru itself.

Any other places (sunrise/sunset, Kata-Tjuta, etc) requires either a tour or using a rental car. Cars sell out quickly so act quickly since you're traveling soon. The car companies at AYQ are Budget, Avis, and Thrifty; we got the best deal from Thrifty in early July. Go through their Australian websites; also check www.lastminute.com.au and similar sites.

(Of course you could sleep in the rental car, solving the lodging problem)