Must do Salt Lick BBQ in Driftwood. It's a 25 minute drive, but so much better than the satellite store in Round Rock. They do all the cooking in Driftwood and transport it to the other store. Best to get it straight from the pit. Another good BBQ spot is Rudys. Get the creamed corn and Cutter's Brisket.
I heard that NXNW restaurant closed which is a bummer. Really amazing food.
Go to Stubbs if there is music playing, but skip the BBQ. I find it to be touristy, overcrowded, long waits, and awful BBQ.
Out East (past I-35) on 2nd, 3rd, and 4th is some really gritty neat stuff... food trucks, a laundromat turned into a bistro, good unpretentious Austin food.
Do a dinner or drinks out at the Oasis on Lake Travis. It's way up above the lake and try to catch a sunset.
Saltgrass is a lovely restaurant. It's a chain, but at least it's a TX local thing. Great steaks. In a similar vein, La Madeleine is a cafe-style french place that has awesome grub. Same goes for Chuy's.
Barton Springs at Zilker Park is a neat thing to see. Natural springs that is also a public pool. Very neat little walk-through museum/science thing that gives you all kinds of good info on the geography of the area. You'll feel almost like you're in PA. Remarkably similar geology. Zilker Botanical Gardens are also lovely.
Check for Music playing at the Belmont. If you want an evening out for fun drinks, I suggest West 6th. 6th Street is the (sorta) Bourbon St of Austin. East 6th is where all the college kids go to drink til they vomit. West 6th is more of a 30s-50s crowd. 5th and 4th have some neat bars on the west side as well.
You should be there when the bats are around. South Congress bridge is home to the largest colony of brown bats and at dusk tens of thousands all take flight. It's a sight to behold.