Not trying to be funny, but there are seriously no good places in Gainesville. That place is truly awful. There are no redeeming qualities about it in my opinion, other than the pretty trees. After we win, I wouldn't imagine it will be safe anywhere in town to be young, tipsy and celebrating a Vol win. UF is filled with carpetbaggers and Superbros. You will think you just stepped out in Ocean City MD, Philly, Atlantic City or south beach more so than somewhere in the South. Southern hospitality got lost somewhere in the panhandle of Florida, and left the rest of the godforsaken state to the Yankee invaders. If I were you, I'd set up a tailgate and just hang out there after the game with just your UT brethren. Going to bars afterwards is just asking for unnecessary trouble.
All that said, after the 2011 game, my friends and I (about 8 of us in the same age range as you are now) went to the worlds worst gentleman's club somewhere near Gainesville. They didn't serve alcohol, they didn't turn down the lights, and the girls (actually, old haggard women) danced on a lunch counter while truckers ate club sandwiches underneath them. Picture a waffle house that somehow had been turned into a gentleman's club. Needless to say, that was a miserable trip.