Swim in Barton Springs during your first few weeks here. The water’s 68 degrees all the time. Nice when it’s at least 95 out, but downright chilly for the rest of the year.
For barbecue, check out the Salt Lick in Driftwood, but be aware it’s BYOB and it doesn’t take credit cards. Take Mopac/Loop 1 south 'til it ends, turn left, and drive 5(?- less than 10, anyway) miles 'til you get there. On weekend nights there’s a 1-2 hr wait. That’s where the cooler of beer you brought comes in.
For other barbecue, check out the Iron Works near downtown. Other good places to eat would be Chuy’s (Tex-Mex, very Austin-y kitsch-y), the Hula Hut (Tex-Mex mixed with tropical fare), Hoover’s (soul food/home cooking), Kerbey Lane and the Magnolia Cafe (super-casual eats), Eastside Cafe (nice, but in an Austin-casual way), and Z Tejas (same).
Also find out about the Austin Hot Sauce Festival, held in August every year. The aforementioned Austin Chronicle is its main sponsor. Always fun. Always really hot.
Besides Enchanted Rock (actually a bit past Fredericksburg), check out Pedernales Falls State Park (on the way there). For longer road trips, look into South Llano River, Lost Maples, and Garner State Parks, all 3-4 hour drives to the southwest in the heart of the Hill Country. During the summer months, try inner-tube float trips along one of the rivers around San Marcos and New Braunfels, or at Garner or South Llano River SP’s. (South Llano’s float is just a short bend in the river.)
Not sure what’s affordable for you, but by Austin standards, the area roughly between Mopac/Loop 1 and Brodie Lane and south of William Cannon Blvd. is fairly reasonably-priced ($150-200K) for being as close to downtown as it is. It’s not manicured suburbia but it’s not a scary bad neighborhood, either, and it’s on the south-to-southwest side. For nice, more upscale manicured suburbia in this direction, go a bit further west to the Circle C development.
See you soon!