Have you checked out our
Offroad / 4x4 / ATV Web Guide for links to clubs and resources? They might guide you to some possible riding locales.
Unfortunately, we have a marked lack of ATV trails in the state, which seems a little paradoxical considering Texas' ranching history and the prevalence of ATVs still used on working ranches and the like.
I'll actually have to dig around before I can provide you with any information. My first thought was Padre Island National Seashore, but that would be false as the National Park Service only allows ATV use by their park staff and, with pre-approval, some volunteers during organized beach cleanups or "Turtle Patrols" (to spot nesting sea turtles/eggs).
So the only areas that come to mind are probably some of the National Forest lands in East Texas and, perhaps, some of the far West Texas lands, though since most of those are National Parks (Big Bend, Guadalupe Mountains), you're probably out of luck with ATVs there, as well.
Austin has a cool mountain bike and motocross park, but I don't think ATVs (3- or 4-wheelers) are permitted there -- only dirt bikes.
I'll post a follow-up after I've poked around a little bit to find more info. for you.