I am an Austin native and here is my favorite restaurant in town.
Mexican
Fonda San Miguel
This is true regional Mexican food, with representations from almost all the regions on the menu. It has been in business for over 30 years, is upscale and reservations are highly recommended unless you like to wait. Dishes to eat: Ceviche, queso con rajas, anything with quitlacoche, mole poblano, camarones in salsa verde, chile relleno, there homemade tortillas, crepas con cajeta, margarita with pear brandy!
They have the most spectacular brunch in town where you will find Chef Miguel Ravago serving you your brunch in the middle of the room, reservations are needed, however you can sometimes get a seat at the bar or the cocktail lounge. For the best service ask to sit in Tom or Jennifer's section, though all of the staff is knowledgeable and professional.