local races usually don't have much in the way of requirements for beginner racers. You do have to be riding a standard road bicycle with drop handlebars and you'll need to at least purchase a 1 day license - usually available at the races for beginners, but to race cat 5 (that's the category for beginners) that's about all you have to do. I would suggest you get some experience riding in groups on your road bike when you get it, before you try a mass start race, but time trials (where each rider goes off one at a time and races against the clock) are pretty much safe even for complete newbies. Many areas have training races that are accompanied by beginners clinics. If you do start racing unattached (meaning you aren't a member of a racing club) do only wear a plain jersey and shorts. Few things looks sillier than showing up to your local race wearing some reproduction pro kit...
To find races in your area see if your state has a bicycle racing association - usually you'll find a web site with a calendar of races (check out Washington's at
www.wsbaracing.com) You may be able to find your local association by searching
www.usacycling.org - you'll also find a lot of information about racing there.
I'm guessing you are using the word race pretty loosely..... you mean that you've gone faster than some guys riding road bikes when you've been out at the same time? Perhaps you mean you've done an organized bike ride like a club century ride faster? You certainly wouldn't have been allowed to join an actual road race on a mountain bike.... Just be aware that a real race is very different, but you'll never know until you try.