If you follow
City Park Road all the way down, there is camping along Lake Austin. I don't recall the fee but I think it is under $10 per day. They have primitive sights as well as RV sights. However, it does get filled pretty quickly at times.
The map link doesn't show the road ending in Lake Austin.....but it does. To get to the campsites stay on CP Rd (instead of turning left to Emma)

Doesn't cost anything to ride the trails.......word is.........if you can ride City Park.......you can ride anything in Texas.