It depends. However, I'm not really a fan of "I'm a fan of UT football and UNC basketball" or something of that sort. Sorry, just not buying.
Like with my dad's side, they are from Michigan. They have lived in Michigan forever. The whole family is Michigan fans. I like Michigan, watch their games, slightly read their boards, and always cheer for them. Except in the Citrus Bowl 2002 and NCAA Tournament 2011.
It's Tennessee first and foremost in college. I love Michigan, but that's how it is. Orange runs in my blood.
In pro, I've been a Braves fan since I was very young. They (and I might killed for this) are my favorite team to watch and follow, and that includes Tennessee athletics.
I don't have a favorite NFL team. I honestly don't. If I did, I guess it'd be the Colts since I like Peyton Manning.
NBA, it's strictly my favorite players. LeBron James, Deron Williams, Blake Griffin. I don't follow one team (although, and this might be bandwaggonish, I am starting to like the Grizzlies).