I guess it depends on your scene. IIRC, you’re in your young 20’s. Most people that age like to get falling-down drunk at a bar, drinking fizzy yellow beer and listening to covers of “Come as you are”. If that’s your thing, go to Sixth Street.
Actually, even if it’s not your thing, go to Sixth Street anyway. Cross it off the bucket list. (I tried it once I didn’t like it. Didn’t inhale.)
Also, if it’s in season, go see the bats fly out of the cave. You’ll have to be on the bridge on Congress Ave at sunset.
Speaking of Congress, you can also tour the Texas Capitol, if that’s your thing. It’s the biggest Capitol building in the US, aside from the federal Capitol.
Franklin Barbecue is supposed to be the best in Texas, which makes it the best in the US, which makes it the best in the world. Sorry Sweep, but it’s true. (It opened after I moved away, so I’ve never been there.)
Or you can make the 45 minute drive to Lockhart and eat at the three BBQ restaurants there (Blacks, Kreuz Market, and Smitty’s). If you do that, I highly recommend building up an appetite. And take your own BBQ sauce, because some of those places don’t have any. They refuse to have it in the restaurant.