I did a motorcycle trip to Kerrville last June and spent a few days there. Other than the main highways going through, there's not much in the way of secondary roads. You'd be at the mercy of Farm to Market roads in the surrounding area.
Look at Google Maps and hit the "bicycling" option and you'll see a very short trail near the river. Otherwise, you've just got major roads and a sprinkling of residential streets.
I also like to drag the little yellow Google Maps man down and see a streetview of different side roads to get a good judge of bicycle-ability.

Here's my trip album if that helps. Wonderful area for motorcycling though!
Hill Country Vacation