You will need various registrations and qualifications to participate in performance events. Use the contact information for each activity listed here for the authoritative information, but generally it looks like this:
Apply for LGRA number for your dog, sending in your dog's AKC registration and the other information requested on the form. Once you've done that, you simply attend an LGRA event, do a couple test runs to convince the club that your dog is safe with other dogs, and you're ready to go.
Apply for a NOTRA number for your dog, sending in his AKC registration and contact information. At that point, you're ready to go!
There is no ASFA number for lure coursing. What you do is register to attend an event, letting the organizer know you'll be doing a "certification run," which is to have your dog run the course to ensure he doesn't interfere with the other dog. This certification is the only thing he'll be able to do at that first ASFA event, and you must submit a copy of the signed certification form back to ASFA. At that point, you're ready to go.
Similar to ASFA registration, but rather than a certification run, you'll need to run two "JC Tests" before your dog is qualified to enter regular trials.