"Best" is so subjective. Also, as others have stated, it depends on your location for most rivalries. The exception is Army vs. Navy. I also like Army vs. Navy because while it is heated, there isn't any real nastiness behind it as seems to run through some of the regional rivalries.

Let me use Florida vs. Florida State as an example. All of my cousins went to either Florida or Florida State (my sister and I (and our mother) are the exception for the women in the family who are college educated) and some of them hold it against the other side and think less of them due to where they went to school, which to me is absolutely ridiculous .

Of course, a favorite slogan of mine is "I root for 2 teams, Boston College and anyone playing Notre Dame". For those who do not know, BC and Notre Dame are the only Catholic colleges/universities which have Div. 1-A football programs.