You have 2 real choices, firstly its unlikely you will spotted and get a proper team driver since they only look at kids.
1st choice, buy a cheap mx5 or something and race in a regional championship, I don't know much about this option, it can be quite cheap however has the potential to be very costly, think cost of atleast $10,000 a year. This option lets you race at larger curcuits.
2nd choice, this is what I'm alot more knowledgeable about. Karting. You can either do sprint races driving single speed karts at smaller tracks, this is what I do, and is lots of fun, really close racing. Costs are about $5,000 a year to be competitive. Or you could do gearbox racing, this allows you to race at both short and long (well known) curcuits, these are seriously quick, and can be quite cheap if you have some mechanical knowledge at about the same cost as sprint curcuits.
My advice would be to go down the karting option, if you want to speak to me more about it, PM me or look over the karting thread on here (its on page 1, 2 or 3 of the motorsports section. If you have any questions then don't be affraid to ask.