I second Salt Lick BBQ ; but Franklins BBQ is AMAZING, and well worth the wait ( you pretty much need to queue up at around 9 in the morning to get lunch there). Try Flip Happy Crepes if you're a food network fan (they beat Bobby Flay in a throwdown a few years ago).
For breakfast food, Magnolia Cafe (it may sound odd, but try the Tofu chocolate cheesecake - delicious regardless of how you feel about Tofu), and Kerbey Lane Cafe - Original are institutions; but Max's Wine Dive is a more recently added "instant classic". For something FAR up market, give Green Pastures Restaurant Sunday Brunch a try - their milk punch is a unique and tasty local tradition.
For Austin style Tex-Mex, all day eats and late night drinks, the classic mainstays are Trudy's and Chuy's Restaurant (made infamous by Jenna Bush's underage drinking charge). They both compete for Austinite's hearts with their individual takes on a local tradition called the "Mexican Martini". Try both versions, decide which you prefer. Lots of good food at both, but if you want something comment worthy in a deep fried, heart stopping sort of way try each one's take on the deep fried, stuffed avocado.
Go to Frank Restaurant for hot dogs. They also have excellent coffee here.
And this isn't so much a restaurant as it is a hot sauce shop, but you need to stop by Tears of Joy Hot Sauce Shop. Try "Austin in August" or one of their other homemade hot sauces!
Dine at the counter at Foreign & Domestic Food & Drink, an updated version of a mid-century diner. Food is anything but ordinary-- the beef dishes are spectacular, but the roasted acorn squash curry proves that vegetarian dishes can pack a lot of punch. Order the yogurt with dill syrup and cherries for dessert. Sounds weird, but it’s spot on.
For some BBQ, JMueller BBQ is a new contender for some of the best 'cue in town. Go early-- once it's gone, it's gone.
For some food truck action, East Side King is a cash-only food truck serving inventive versions of Asian classics. Think steamed buns filled with avocado or crispy chicken skins with kimchee.
There's also a lot of great Mexican food in town. In a city full of great Mexican places, La Condesa is my pick for something a little less down and dirt. I love Chef Ortiz's fantastic, modern takes on guac, tacos, plus killer drinks & desserts (pastry chef Laura Sawicki is one of the best in the country).
Another good one is Tacodeli. Way casual taco spot with some serious food. 30+ tacos, plus sides and amazing salsas. Order a few tacos, but be sure to try the puerco borracho, made with slow-roasted, succulent pork braised in tequila & sherry and seasoned with fresh basil and pasilla pepper sauté. Yum.
To satisfy your sweet tooth, the sticky bun at Olivia is warm and sweet. Perfect.
Do NOT miss Rudy's for barbeque! Texas was amazing in that every restaurant I went to had portion sizes to feed six people.