I now live in Austin, TX but in 2007 we came here on vacation. Some of the highlights we enjoyed and have revisited.
River Walk IMO is bleh, just a bunch of shops/bars/restaurants BUT the River Cruise is worth it IMO, its fun and you learn a lot of the local oddities in regards to the River Walk and architecture.
The Alamo of course, then take a walk to the St. Francis Cathedral where the remains that were found in the floor of the Alamo are entombed.
New Braunfels:
Schlitterbahn Water Park - this is the most amazing place if you like waterparks
Natural Bridges Caverns - beautiful wet caverns
Snake Farm - interesting road side attraction
Boerne:
Cave Without a name - this cavern is fascinating as its a mix of a wet and a dry cavern.
Lockhart:
Home to several famous BBQ places. My favorite is Blacks
Dripping Springs:
As someone else mentioned, Saltlick. The original in Dripping Springs is all about the atmosphere, bring a cooler of beer, kick back, relax and enjoy. Personally, while I really enjoy the food its not the best BBQ in the area IMO.
San Marcos:
Aquarena Springs, complete with the glass bottom boat is an interesting adventure.
Fredericksburg:
Old Tunnel State Park - well worth the $5 during bat season to watch the nightly emergence (seasonal and bring LOTS of bug spray)
Old town with shops/restaurants is a fun day of shopping
Austin:
I went to the botanical gardens for the first time last week, very nice!
Lots of things: The Capital Tour is informative, the Texas State Museum is full of all things Texas, LBJ Library previously mentioned, Congress Ave. Bridge (bat central here in Austin) "SoCo" South Congress Ave home to Big Top Candy Shop and Uncommon Objects along with several local eateries. Lady Bird Lake, Lake Austin and Lake Travis. Zilker Park, Barton Springs Pool
Off the beaten path a bit: Hamilton Pool (need reservations for at least the weekends)
Marble Falls:
Longhorn Caverns - these are "dry caverns" also very unique and not physically strenuous.