Most of the good places I know of have been touched on in posts above. Some additional detail on some:
Pedernales Falls SP. You can get 17 miles out of this park if you do all the trails including across the river, Jones Spring/Wolf Ridge, and the horse trail which is not on the park map but the rangers will give you a map of it if you ask. There's also a really nice bird blind visited by many species in winter. The park is especially nice when it's been raining and the creeks and springs are running.
Enchanted Rock is kind of small for hiking--you can get about 8 miles out of it, but the scenery is spectacular. Good place for contemplation, photography, and nature study. We used to do Thanksgiving with all the trimmings at the campground. Call early for reservations for this park. A great place to take kids on their first backpacking trips.
Bastrop State Park, just east of Austin, has an 8 mile loop through piney woods with gullies, rock outcrops, hidden ponds, and, just for fun, about 100 permanent orienteering markers (with map) that you can use to practice your map & compass skills in deliciously confusing terrain.
Lake Georgetown is a pretty area but has a very frontcountry feel to it since boaters can drive their boats up and party anywhere. But all the way around is 16 miles. Scouts use it for practice.
Guadalupe Mountains National Park is spectacular. And challenging. No water to speak of up on the trails. I've been to the Rockies, California, Pacific NW, Smokies, Glacier, Ozarks, Big Bend, Whites in NH, and some other miscellaneous places, but there is nothing quite like the upper reaches of Guadalupe. You just have to go see for yourself. It is a long way from anywhere, though. Not a weekend destination. Not necessary to wait for the cool season; it is high enough in elevation to be comfortable even in August. A good opportunity to visit Carlsbad while you are over there.
Big Bend has a whole lot more desert hiking than mountain hiking. It is a vast region of desert hiking, with a spot of mountains in the middle where all the people (including me) go. Also a day's drive from Austin, farther from Dallas. There is also Big Bend Ranch State Park next door, another vast region of desert and hills, almost entirely undeveloped. I haven't been there but some friends enjoyed it very much.
Lost Maples is about 3.5 hours from Austin (winding roads) and has only about 7 miles (maybe more soon, they just bought a bit more land) of trails but they are truly excellent miles with deep canyons, ridgetops, streams, springs, and barking frogs to keep you company at night.
On any trip going west on IH-10 from central Texas I strongly recommend stopping at The Caverns of Sonora. Privately owned, 2 hour cave tour, $20/person, and utterly enchanting, really. One of the most "decorated" crystalline caves anywhere. Nice proprietors, car camping.
From Dallas +1 on the Ozarks. Not like the scale of the west but you can find some pretty rough country in there, together with lots of history. And, unlike Texas, a decent amount of public land.
Other posters are right about the water, though. Most places in central or west Texas, get used to carrying it.