Getting some track time at a good driving school like Skip Barber or Car Guys is a great start. They will teach you alot of the theory of handling dynamics, as well as critique your driving as you go. but that's just the tip of the iceberg if your really thinking about going racing.
I'd suggest entering some local autocross competitions first. They are solo runs on a course marked out with cones in a big parking lot. Speeds are usually under 60 mph (courses are tight), street cars can be competitve, and the price of a mistake is usually a spinout on flat pavement rather than on aff-course excursion. All you usually need is a car, $40, and a helmet (and you can even borrow that).
The participants are usually VERY willing to help newbies through and give pointers. Autocross gives you a lot of experience at handling the car at its limits.
Assuming you go to track days/school, you will need alot of time and money if you are interested in amateur road racing competition. You will need an SCCA competition license (which is sometimes available from schools). Typically you would begin in Showroom Stock, where cars are (mostly) stock with the addition of a roll cage. you definitely DO NOT want to race your street car. A good used SSC-prepped Neon, Protege, or Civic is going to run you $5000-10,000, probably with some spare parts and tires included. A race weekend will cost a few hundred dollars for tires and entry fees, plus transportation and lodging. You'll need a friend or two to crew for you. You'll need access to a tow vehicle. You'll need mechanical ability or access to a mechanic. You'll need to be able to stay up late most nights during the season, getting ready for the weekend, then you'll need to get there by early Saturday morning for registration and practice.
All of this comes out of your own pocket. Your insurance will not cover a race car. Nobody is going to finance your startup, or give you a ride to begin with. Once you have some successes on your own, you can bring a package to potential sponsors, showing what you can offer them. Remember, they don't need you; you need them. The beginning is all sweat, time, and money put in by YOU. Don't believe a driving school ad that says they will make you a "real" racecar driver.
Heheheh, I could go on. I don't race, myself, for all of the above reasons. However, I have a number of friends who do, and I have been volunteer crew on occasion, so I know how hard they work, how much they spend, and that they don't get much except satisfaction from participating. But just like anything else, if you have the inclination, that's enough reward!
