If you have changed your mind to a getting a dirt bike I would get a 250 4stroke. I don't know much about quads (atvs) but I assume around a 400-450 would be a good size to start with. I personally would go with a bike.
Save your money, find some other people that ride dirt bikes and learn to ride. Worry about the racing later. Learn to ride (bike or quad) well and learn how to work on it. You at least need to know the basics, oil change, air filter cleaning, tire change, brake change, and any other maintenance. Knowing these things will save you a lot of money over the years.
A desert race is not the place for a new rider. Everyone has to start somewhere, so start by trail riding. Find some people to ride with. Get comfortable on your bike. Push to get faster and then see if you still have a desire to race. There are many, many more people that have tons of fun just our play riding than those that race.
Write down your goals and look at it every day. Put the list on your bathroom mirror. Don't lose sight of your goals. Save your money! And when you have enough money for a bike and proper riding gear, get started. Get out and ride that bike as often as you can. When it breaks, learn to fix it. And then ride it some more. There is no replacement for seat time in my opinion. For about 6 months back in my racing days I lived where I could ride right out of the house. I improved my riding a lot because I rode almost every day. Sometimes only 30 minutes, but other days for several hours. I know that isn't possible for everyone, but if you want to race, you need to learn to ride well, and seat time is the best way to learn that!
Good luck!