Fenway Park (above)
The oldest major league stadium has more character than most. It opened in 1912, and has been packing in fans ever since. The Red Sox have sold out every home game for over 600 games and counting - a major league record. Fenway is a legendary place that every sports fan needs to visit at least once. And since they have won two World Series in the last decade, I do not even need to mention the pesky curse.
Getting there and tickets: Located in Boston, Fenway Park is right off the Mass Turnpike in the heart of the city. Take the green line on the T to Kenmore station to reach the ballpark. Tickets can be purchased here. Since games are always sold out, it is easy to just pick scalp your tickets or grab a pair off of craigslist, ebay, or stubhub.