Find a driving school you want to attend. Take driving classes. If you can control your car well, and know the physics of things, racing will become a lot easier. Take normal driving classes first, I used Skip Barber Hi Performance Driving 2 day course, and I can now go to the race classes if I want. The driving schools give you some very necessary skills that you will definitely want to learn.
Tracks normally have open track days wher eyou can just pay a small fee and race your car all day (Hotlapping. Sometimes there are rules like no passing in certain areas etc. etc.
Same thing with drag races. you can go to your local drag strip (my closest one is sears point) and they have test and tune days. Sacramento raceway has weekly races where you bring your car and race it there. The only thing that you can do is practice practice practice. For dragging, at first, just start slow. Take off like you would normally a few times, then progressively let the clutch out faster and faster with more rpms and you'll learn to launch it. ALWAYS DO THIS IN A CONTROLLED ENVIRONMENT and not on the street.
Racing just comes down to practice, knowledge of how cars handle, and knowledge of the limits of your car's setup at the time of the race/drive.
Links to get you started are below, I just looked and found a few places that you should check out and read up on. They might have more info on how to get into racing.
www.skipbarber.com (The one I went to, can't say enough good things about it, if I could afford it, I'd go back in a FLASH)
www.bondurant.com (another one I know is very popular, I haven't much research on it)
http://www.sportscardrivingexperience.com (This looks to be very commercialized but probably gives you some good knowledge and it's 1 on 1 with an instructor)