The best actual hiking in Austin is found on the Barton Creek Greenbelt. There are numerous entrances to the greenbelt, which one you choose will depend on what section of the greenbelt you're interested in. It goes all the way from Zilker park, just south of the river and accessible via Barton Springs Rd, all the way down past 360. If you go far enough south, you will reach water in Barton Creek (most of the creek is currently dry this year).
For the more casual hiker, you could also check out the Hike and Bike Trail, which is on Ladybird Lake (formerly Town Lake). There are a lot of different places to park; the easiest in terms of always finding a spot is underneath Mopac on the south side of the river. Get there by taking Stratford drive north of Barton Springs Rd and turn into the gravel parking area underneath the highway. The hike and bike trail is perfect for people watching. The most active portion of this trail is on the south side of the river, between Lamar and Congress, where you'll find a large dog park and wonderful views of downtown, as well as a statue of Stevie Ray Vaughn.
Lastly, also for the casual hiker, is Mt. Bonnel. This is a short walk up a bunch of stairs, or a short hike up a rocky trail. From the top of the "mountain" you get a lovely view of hill country in one direction; in the other direction, a nice view of downtown. The park is located on Mt. Bonnel Rd about halfway between 2222 and 35th St. If you're coming from 2222, it's off the stretch that's between Mopac and 360.
Note that Austin is largely flat, so there won't be much elevation change on any of the hikes you'll find here. However, the terrain is quite rocky on the Barton Creek Greenbelt. Nevertheless, there aren't any hikes in Austin that couldn't be managed in flip-flops.
Hope this helps.
