There are a few different sports that run throughout the summer months in the US, it just depends what sort of thing you're after.
Soccer/Football, not to be confused with American Football (NFL). The MLS (Major League Soccer) season runs throughout the summer in the lead up to finals in December, so the summer months will see the season well under way.
As the sport grows in popularity it is beginning to attract bigger and bigger stars, and recent seasons have seen performance by Thierry Henry, Tim Cahill and Robbie Keane.
Baseball is another good option, although I wouldn't advise it unless you are familiar with all the rules and are prepared for a game that can last as long as 3 or 4 hours!
Tickets aren't cheap either, and you'll be looking at over $50 to watch big teams like the Boston Red Sox or the NY Yankees. For a quintessential American sports experience through, you can't beat it.