Registration most events is done via BikeReg these days. You can use their CX Event Calendar to find events near you.
Racing is divided into "Categories", or "Cats" for short. Beginner women start in Cat 4 and beginner men start in Cat 5. As you progress you move "up" in cats going into smaller numbers. So as beginner once you win your first race, or complete 10 races, you're eligible to cat up into Cat 3 for women or Cat 4 men. After that the rules for upgrading are a bit more complex.
Some events choose to run a letter based system, like A, B, C, etc... where generally the 'A' group would for the elites/cat 1s and 2s or the people that consider themselves "fast", and the B and C groups for those of lower categories. When in doubt, ask!
Races are time based, so the number of laps you do depends on the speed of the faster riders and course conditions. Beginner races are typically 30 minutes, so if the first lap or two are completed in 6 minutes you'll end up doing 5 laps total. The "A" and Cat 1/2 racers can go for up to 60 minutes.
If an event is USA Cycling Sanctioned you'll need an active annual "race membership" with USAC or buy a day license. You can buy a one day license when you're registering for an event via BikeReg. You can get your annual license/membership at https://memberships.usacycling.org/.
If you have an annual, make sure to have a copy of it on your phone/pulled up on the USAC website when you check-in to race!
Also checkout our Intro to CX Doc if you haven't already!