Cool topic.
I'm originally from western Connecticut, an area that's in both the Red Sox and Yankees areas. (Mets too, but not as much)
Moved to western Mass, just in time to discover sports, 73 world series, Aaron setting the record, Rice and Lynn and the whole 75 season and series. Dad is also a Red sox fan, so I guess it's natural. We moved near Boston just in time for the 78 season, and at the time you could just get on the T - Bus stop was less than a mile from home - and go in for a game. There were usually tickets available if you weren't too picky about seats, and a high school kid could afford an occasional game.
Of course I hate the Yankees. It's sort of required.
For the better fans, the rivalry is just that. The Yankees have had a load of great teams, and back then were usually just a bit better late in the year. Lots of great debates over the years, Fisk or Munson? (They were pretty evenly matched, Munson was a bit more durable. I really wish we'd gotten to compare them both over a long career. ) Nomar or Jeter - (In his prime Nomar was a bit better I think, but wasn't the same after a broken wrist and faded pretty fast. Jeter is obviously the better of the two, and while I rarely bought the hype that came with some of his great plays he was just too good to actually hate. )
Since I like the crazier aspects of baseball that are sadly gone now, I always liked the Brewers home run slide. Not the new one, but the old beer barrel and mug.
Football it's been the patriots since the early 80's. They usually looked like they'd be a good team but just weren't. Lately though.... at least now I know how Yankee fans feel.
I don't really follow Basketball or Hockey much anymore. But I was a Celtics/bruins fan, and like those teams most of the time.
In all sports I don't really dislike most teams. I've had some favorites that have come and gone, more certain players I liked than teams.