My sweetie and I visited Austin over the Christmas hols. We really like Lambert's for "fancy bbq." Lambert's seems to have mixed reviews on the board, but thought it was a very tasty... especially liked the brisket, dry ribs, and the cheese/charcuterie app (Texan cheese is fab!). Also, it's a nice place to sit and hang out for the night (with music, although you might request to sit downstairs if you'd rather not be in the same space as the band/musicians).
We liked Starlite (2nd Street warehouse district) for a causal/romantic evening... superlative veal chop, good cocktails... note their happy hour specials on drinks and apps.
And we had a good Italian meal at La Traviata... fun a la carte antipasti menu (Texan goat cheese), refined pasta dishes... and a very pretty room.
Please do post your own finds - we'll be back to Austin and I'm still collecting data for our next trip.
If you have a car, sunset at the Oasis is beautiful, but the food is absolute crap (have a marg and some chips and get outta there with the sun).
If you are looking for a fun atmosphere then Chuy's, Magnolia, Salt Lick and Kerbey will treat you well. However, if good chow is your main objective then these places will leave you very disappointed. Each of these is as close to a chain as you can get without being a national chain.
I'll suggest some alternatives that I believe have better food:
BBQ- in Austin try Sam's or House Park; even better is to head out to Lockhart
Tex-Mex- Habenero Cafe for breakfast or lunch; Angie's for carnitas tacos (not the carnitas plate) for lunch
Tex-Mex and Interior Mexican- Conchinita pibil at Azul Tequila; Curra's on Oltorf can be hit or miss
Burger- Casino El Camino
Southern Cooking and/or Soulfood: Ben's Longbrach; Galloway's Sandwich Shop; Gene's Po Boys; Tony's Southern Comfort
Upscale: Uchi for sushi
Have fun and please report back.
The most important thing is to get a couple tacos out of a good taco cart or taqueria, preferably one with hand made tortillas...check out the Austin board for recs. I guess you could go to Kerby lane for the white queso dip, migas, and "Austin Vibe". I wouldn't call it a must eat tho...course I don't think there is such a thing..
One of my favorite places in Austin is Vespaio and their "cafe" next door, Enotecca Vespaio.
We are also big fans of the Magnolia, which has several locations around town - the South Congress one is our fave. They have faaaaabulous breakfasts. I've never had a bad meal there. They have great sandwiches and their Mexican food is pretty good as well - not the same as a traditional Mexican restaurant, mind you, but good that strikes your fancy while your sweetie wants a burger.
There is an Austin thread on Chowhound that you'd do well to look at.