You can go to AKC website and find information of all the competitive event they have to offer.
www.akc.org
To get started, I would recommend finding a good dog club or training school. In some case the dog club is the training school. A good thing to start off learning is obedience.
Then afterward it really depends on you and what you want to do. There are other organization that hold dog events as well like USDAA-agility and NAFA-flyball.