For places easily reachable by bus or inexpensive cab ride:
Esther’s Follies on Sixth Street is an Austin institution offering live sketch comedy and magic. Lots of fun. You’ll probably need to reserve your tickets in advance. Combine it with a stroll up and down the length of Sixth Street, just for fun people-watching.
Zilker Park, just across Lady Bird Lake (actually a river) from downtown, is a terrific urban park. You can swim in Barton Springs, although the 68-degree water makes for a cold swim. Great hike-bike trails, nice botanical gardens.
Food-wise, barbecue and Tex-Mex are the quintessential local cuisines. For centrally-located barbecue, Franklin’s is probably the way to go. The original Chuy’s on Barton Springs Road (right near Zilker Park) is decent and certainly iconic Tex-Mex. Other good places not far from where you’ll be staying are Hut’s Hamburgers (Sixth Street a few blocks west of downtown), Hoover’s (home cooking, on Manor Road a few blocks east of I-35), and the Eastside Cafe (just good food, across from Hoover’s).
Bar-wise, the Warehouse District (centered around Fourth Street, west of Congress) has several good bars that cater more to the 30-and-up set. I also rather like The Tavern (12th and Lamar), although it’s more of a bar-restaurant hybrid.
If you do history stuff, the Capitol, the Bob Bullock Museum of Texas History just north of it, and the LBJ Library and the Harry Ransom Center at UT have a lot of interesting things going on- all are cool.
If you’re willing to rent a car for a day or pay for more expensive cab fare, Mount Bonnell provides a beautiful view of the city to the east, Lake Austin (also a river) directly below, and the hill country to the west. Terrific place to watch a sunset. As is the Oasis at Lake Travis, if you want to have a drink or a so-so dinner while doing so.
Enjoy your visit!