Agreed heartily with Michael L's list of places and seasons. Of particular note, Enchanted Rock becomes a truly enchanted wonderland after a good storm. I've made it a point in the last few years to get out there if I ever see big weather coming to the region over the weekend. The whole granite scene comes alive with flowing water through every ravine and fissure in the monolith -- it's a sight to behold.
Basically if you want to get away for a quick night or two, you're limited to backpacking in the small-ish state parks and natural areas that are within 1-3 hours drive from Austin. Most good things are west of Austin, since going east quickly takes you into a rather boring and long coastal plain(ish) that stretches on down to the gulf of mexico. East-southeast, I guess you'd say.
Of those little state parks within weekend-shot of Austin, my favorites are consistently Lost Maples, Enchanted Rock, and Hill Country State Natural Area. Generally, unfortunately, you'll be required to bring your own water on whatever trip you take through those areas. But hey, it's just a night or two, and you can't hike more than 5-10 miles through those places.
Further afield of Austin, there are good opportunities in: Big Bend, Guadalupe Mountains, and Snake River. The lattermost, I've never visited, but I've at least heard promising things about it. Big Bend is, I might hazard, the best backpacking in the state, and it will take you at least 2-3 trips to really exhaust the main route possibilities there, which is great. If you're hardy and don't mind carrying a lot of water, you could probably spend a couple of years hiking through the less-traveled desert routes out there.
Finally, as Eugene said earlier -- when I want to do some real backpacking, I get on over to the lower rockies in NM. The drive is somewhere between 10 and 14 hours, depending on where exactly you're headed. Getting up the interstate to Colorado doesn't take much longer than that, once you're already in NM. The possibilities in those state, of course, are endless. My favorite areas in NM are Pecos and Wheeler Peak, which can both easily be reached in the timeframe given above.
And THEN finally, within Austin, there are numerous DIAD activities that will keep you exploring for years. There are caves to crawl through; there's the waterfront with kayaking, canoing, and stand-up paddleboarding; there's running along about a dozen different trails, from easy crushed granite along the waterfront to the backwoods of Bull Creek; and if you want to challenge yourself with some day-hike training, you can yo-yo the entire Barton Creek Greenbelt for a total distance of 14 miles. Then, after those things ... there are a thousand other things. Rock climbing at about 10 different crags in the Austin area, or three different indoor gyms. A vibrant biking scene with numerous weekly social rides or more serious training groups. The list goes on and on.
Not to mention, Austin is probably overall one of the most hip cities in the whole country. This is somewhat of a recent phenomenon, as the city that I grew up in (in the 80s and 90s) was somewhat undiscovered and still a *little bit* quiet. But now it's roaring and growing and fuming with new fancy restaurants, thousands of live music performances every month, events downtown and in the public spaces every weekend, huge festivals that draw giant crowds from around the country, and just lots of really cool people doing experimental, innovative, progressive things. All while staying relatively chill, un-starched, and playful. It's really a neat town.
The only downside to all this, is it's becoming rather expensive to live anywhere near the center of town. And Austin natives like myself are becoming somewhat of a rarity.
Hope this is helpful :-)