One of the best ways to get into bike racing is to join a local USA Cycling affiliated club. Many clubs help develop new riders and teamwork is an important aspect of racing. There's nothing better than friends in a peloton!
USA Cycling's website http:/www.usacycling.org has a directory of clubs. Not all clubs have open membership however. In getting started, you'll want to find one that accepts members of all abilities.
If you're a college student, check out your school team. There's no better way to get started than in collegiate racing where it is a team sport at its purist.