Best places for burgers, Mexican, and/or BBQ downtown?
For Mexican: Garrido's in the Austin 360 building on 3rd is very good: inexpensive, very tasty, unusual "interior" Mexican cuisine. Go wild and spend the $1.50 for the chips and hot sauce. The hot sauce is very unique, and refills are free.
The last time I was there they had some off-menu specials, including a chile relleno nogada that was exemplary, for about $12.