The local clubs in your area will be your connection to these events. They work very hard to find locations (typically school fields, ranches or parks), coordinate all registrations, and put on the events themselves. These are completely voluntary efforts, nobody's making money off them. In fact it's typical protocol to assume that every attendee should be out helping in some way at the event when they're not running their dog. Lunch is usually pot-luck, and you'll get to know folks quickly. You might start out by attending some events before you even bring your dog, to get to know the people and how the event works.
As mentioned earlier, you might find several clubs in your area that run events. As these clubs as sanctioned by the parent organizations, you should find all rules essentially the same between clubs, and you will find it's quite easy to attend an event with a new club as a result. Keep in mind that these events will take entries from all sighthounds, and even some non-sighthound breeds. So you do not need to find a club called "Borzoi Racing Club of Yourtown" to get rolling. All you need to do is find any club that's reasonably close to you. Contact them, see where they host events and if there might be any restrictions, get on their mailing list for event announcements, and you're set.
When you find an event you want to attend, you'll need to get the "premium" (event announcement and logistics) which will contain an entry form. Typically these days, the premium will be distributed via email, by the officers of the club. Typically you need to return the entry form and entry fees (usually $10-15 to enter a dog) in advance of the event, though some events will take walk-up registrations as well. If you're entering for the first time, just exchange an email or two with whomever is distributing the premium to make sure they know you're new and they will be glad to answer specific questions about getting started.
So, here are some links to get you started:
LGRA clubs: http://www.lgra.org/info/memberclubs.shtml
NOTRA Other Breed Clubs: http://www.notraracing.org/links.htm
NOTRA Clubs: http://www.notra.org/
ASFA Lure Coursing Clubs http://www.asfa.org/clubs/index.htm
AKC Lure Coursing Clubs: http://www.akc.org/clubs/search/index.cfm?action=perf&display=on
Rally Obedience Clubs:
Agility Clubs: http://www.akc.org/clubs/search/index.cfm?action=agil&display=on