OK I usually never even log in to post anything, but please do not go to f'ing Mandola's at the triangle. It's not awful, but it'd be such a waste of trip to Austin.
I would take somebody to Magnolia's, 100%. Don't even consider Kirby Lane.
East Side Cafe is also very good, though a bit nicer, on Manor rd.
Red River Cafe is my hole in the wall, low key favorite. Austin Java is also a good spot, especially if you're a bit of a health nut - lots of options for gluten free, vegan, etc.
Juan in a Million might not be the best breakfast tacos in town, but it's certainly the largest and most filling. It's a good place to check out.
Torchy's Tacos is pretty good and a good representation of Austin tacos.
A good, Austiny choice is to go to one of the food truck parks (I forget where these are, but there are a few of them around) - outside, good atmosphere, obviously a large selection for anybody's appetite.
For BBQ, the recommendations above are all quite good. If you don't mind a drive, the suggestion for Llano is good and also Lockart has some great BBQ (2 or 3 famous places in that one small town).
I think Salt Lick is good - it's not change your life good, but I love it. The best points about it are the pretty drive out there, the nearby wineries, and the BYOB part is always fun. I think it's a good experience.
For burgers, I'm a big fan of grabbing a to-go bag from P. Terry's and taking it to Zilker Park (good for family). Hut's is good and Sandy's also has a oldschool vibe. A swankier option would be Hopdoddy's - I think they have the best burgers, shakes, and fries in town, hands down. But there could be a wait out the door.
And if you enjoy cafes/coffee/bakeries I would suggest Quack's on 43rd street. Cute place with family vibes and great ice coffee. Thunderbird cafe (on Manor rd.) is a hip place, across from East Side Cafe (see above).
Hope this helps! I live outside the US now, so these are the things I miss about Austin the most and can't wait to have.