1983-'92: New York Mets
The Mets are the team I've gone back to again and again. In fact, I can argue that I never really left them. It all started as a very young boy, just discovering baseball, when my grandfather first had cable installed. One of the channels was a local NY City station, WWOR (where I also later discovered then unknown comedian Mario Cantone's Sunday morning kids show "Steampipe Alley"), which carried the majority of the Mets' games. I spent a lot of time at my grandparents' house, particularly during the summer, while my parents were at work, and subsequently watched a lot of Met games. It's how I learned to love the game.
For whatever reason, newly acquired catcher Gary Carter quickly became my favorite player, although I suspect I may have loved Mookie Wilson nearly as much. Basically, after "The Kid" and Mookie, every other player on the team was tied for my second favorite: Keith Hernandez, "Dr. K" Dwight Gooden, Darryl Strawberry, Lenny Dykstra (despite him often pushing Mookie out of the lineup at CF), Wally Backman, Ron Darling, Sid Fernandez, Howard Johnson--- it all culminated with them winning one of the most exciting World Series of all time, and a young impressionable kid like me was hooked. I fully expected this incredibly talented youthful Mets team to win a few more, and if it weren't for cokeheads Darryl, Keith, and Doc Gooden, maybe they would've.
Anyway, after the brief flirtation with a late season callup of "The Phenom", Gregg Jeffries, and the emergence of David Cone in the 1988 season, that team was swiftly dismantled and soon found itself mired in mediocrity. The "Mackey Sasser Years" of the early '90s weren't all that fun, but I still watched the games in hope that moves such as bringing in Frank Viola, Eddie Murray, and Bobby Bonilla would produce big dividends, but it was never the same. As Chuck Klosterman said about rooting for the Boston Celtics in the '80s, at a certain point you're just rooting for laundry.
Honorable mention: In 1988
Toronto Blue Jays, Pittsburgh Pirates, Atlanta Braves, NY Yankees
Three Rivers Stadium
1988-'95: Oakland Athletics
1999-'05: Boston Red Sox
2008-'10: Baltimore Orioles vs. Washington Nationals