Depends on what/how you like to ride. Walnut is the last place I would recommend, unless you want to deal with dogs running off-leash and pooping everywhere, trails that have the technical difficulty of riding on a sidewalk, and crowds of bikers that think they're racing & run everyone else off the trails. It has a nice pump track and can be fast & flowy, but isn't my cup of tea as it lacks anything remotely challenging (other than the pump track). I guess it isn't bad for beginners or to get a quick spin in during off-peak hours, but isn't a place I would recommend for seeing what Austin really has to offer.
If you want something a little more all-around, the Green Belt is the place to go, as you can stay on the main trail for minor challenges, but can get all the tech you want on the many, many side-trails. I would recommend going with a guide the first few times, as it is a maze of trails. MTBer's in Austin also know City Park, which is pretty gnarly if you're looking for tough stuff that is easy to navigate. The Goodwater Loop at Lake Georgetown is a 27+ mile loop of rock that is fun, but doesn't have much elevation, so it is more of an XC trail than FR--it too is easy to navigate as there are no side-trails. Other good trails you'll need to hit up that have something for everyone are Pace Bend & Reveille Peak, both of which are about an hour from downtown.
I could go on & on as there are tons of trails really close by, but those should get you started & get you a feel for what is around. Enjoy!