Tailgating at UT sucks for the most part, except if you come up on the river and the Vol Navy. Most visiting fans I have seen just find a pay parking lot or get their early enough to find open parking spots on some of the main roads and tailgate there. I usually park at the lot off of Clinch Avenue and Walnut Street in downtown. Parking went up from $10 to $20 down through most of the pay lots, but its about as good as you can do for getting in and out of the crowd. You may find some places to set up off of Neyland Drive SW of the stadium or down by Philip Fulmer Way if you get there early. Also, some people let you park in their yards off 17th street, not sure what they are charging these days, but it usually cheaper than the pay lots.
Instead of tailgating, a lot of people just go down Cumberland Ave. to the strip, lots of bars and places to eat right on campus there.
If you do go out early, directly to the northwest of the stadium, the Vols do their entire pregame (radio broadcasts) and the band marching to the stadium at the hill, which is at the corner of Andy Holt Ave and Stadium Drive.
So the best of the UT game experience is the Vol Navy, the Cumberland strip, and the Hill before a game.