The trails at RiverPlace (subdivision) are nice, although I notice that they charge a fee now.
If you want to do some hiking and primitive camping, you might be able to get a spot at Lake Georgetown. The area is run by the Army Corps of Engineers.
Also, there are nice hiking trails at Milton Reimer's Ranch, a Travis County Park. I did some volunteer trail work there once on "It's My Park Day".
There are also an abundance of City of Austin parks that are very nice. One I like a lot is Commons Ford Ranch Metropolitan Park.