DrC-
There are a couple of options. First, you have club racing that you can do with SCCA and NASA. SCCA has better coverage geographically, features a wide variety of cars, and is probably a little more political. To run SCCA, you will have to buy a race car and do two Competition Schools (usually one in the Spring and one in the Fall for a region, but most guys do back-to-back regions to combine the schools and get it done in a month or so.) NASA is the other big club racing group and focuses more on spec classes (including the spec cobras from FFR). NASA isn't as strong in every part of the country so that poses some limits. One big advantage for NASA is that you can do their HPDEs (schools) with just a street car. There are some philosophical differences between NASA and SCCA when it comes to race licenses: the NASA schools will take you about a year, but you end up as a better driver and you can run up front sooner. With SCCA, you get into a race sooner. Sort of depends on whether your goal is to race or to do well. (Let me go ahead and dispel any thoughts of a win in either group: it takes a little while. There is no such thing as a natural driver - just a guy with more track time.) NASA usually accepts an SCCA license. SCCA also supports a lot more open wheel classes than NASA (in fact, most NASA groups don't run open wheel.)
Another option are the car-specific clubs and regional groups such as BMW CCA, Porsche Club, COM, and others. The licensing requirements are usually substantially easier. Car counts are also lower. In general, these groups are a little more about fun and less about competition. What that means is that usually the drivers aren't quite as good as the traditional racers. (Before the flames begin, I know there are plenty of exceptions. I am talking 'in general'.) You can get a lot of time on track with some of these groups and they are very collegial affairs.
The other major option is Vintage Racing. I've done vintage and consider the term a bit of a misnomer. The better term might be "wealthy guys wanting to drive a car fast but afraid to actually race wheel to wheel'. Vintage racing has some severe restrictions on what you can and cannot do on the track. The idea is to protect the cars. That makes sense - I have seen cars worth as much as $7M on track in a vintage event. So no late braking or aggressive moves on track. Vintage has an even bigger gap between results and talent - you write a check for the fast car in class and more often than not that determines where you finish. In vintage, unless you pay to run up front, you need to pick your racing against cars near you on track and have fun playing. Vintage runs pretty much all around the country with different sanctioning bodies so you can always find a place to run. Vintage licenses also tend to be pretty easy to get, but it does vary.
Finally, there are manufacturer series like Panoz, Star Mazda, and F-2000. The series are run by a car manufacturer, race school, or both. For example, with Panoz you can do their schools, rent their cars to race (so all you do is fly in and race), and then buy your own car and keep racing. Usually geographic specific and you get a lot of coaching and support. Figure $3K'ish per weekend which includes everything but your travel expenses and what you break on the car. Competition levels can vary wildly: one series might have older guy wanting to play while another series (especially the open wheel ones) may have a bunch of 16-18 year olds whose dad's run NASCAR or own car teams and are using the series as a way to move them up to the pro ranks.
In terms of what you will need:
1. The car. It will have to pass basic safety stuff with belts and windows not older than two years in age. You can buy a spec RX-7 for as little as $2,000, but most race cars will be $15-40K. $12-15K can buy a mid-pack spec Miata or A-Sedan (Mustang/Camero) car. FFR spec racers and top Hondas go for mid $20s while some of the faster BMWs and Porsches get into the $30s. Vintage cars can regularly be in the $50-80K range for Mustangs, Jaguars, sports racers, etc.
2. The license. In addition to getting the schools our of the way, you will need a race license physical. Depending on your age, they may want a baseline EEG and to make sure that you can manage your diabetes. Most guys don't have a problem with the physical. License costs about $100 and the physical is about the same at most doctor's offices. Some insurance plans will cover the physical.
3. The hauler. For the NASA schools you can drive a street car for everything up until the final competition license test. For everything else, you are pretty much going to need a trailer. If you are on a budget you can get open trailers for as little as $1500. Single car enclosed trailers can run you $5-8K and gooseneck two car haulers can be $12-20K depending on what you need.
4. Safety gear. You will need a suit, helmet, shoes, and some other things. I would strongly encourage you to wear a head-and-neck restraint systems such as HANS. Figure at least $1K for the basic safety gear and most people will spend a little more than that. Another $650 for the HANS.
5. Track Support. Ironically, most racing at these levels doesn't require a pit crew. Most of the races are sprint races. In any case, you will find that the other racers in the paddock will usually bend over backwards to help you with technical issues and fixing your car. Plenty of guys show up at the track by themselves and do just fine. At the extreme levels in club racing most of the top teams have a helper or two, though. If you are just starting, I wouldn't be too worried. I have won races where I didn't have any help.
6. Off Track Support. Unless you really know your car and race setups really well, you are probably going to need a local shop to help you with some things in between races. Find a local shop you trust and figure out what you need. After every race you will want to do a general "re-bolt" to look for loose things and see what fell off or needs some attention. $500/weekend will get you in the ballpark (starting), unless you have an exotic vintage car that requires specialized knowledge. Some organizations will do the repairs to your car and bring it to the track for you. Just depends on how big of a check you want to write.
7. Spare parts and tools. Varies a lot. Over time you will put together a list of what you want. Figure after a year or two you'll have $5-8K in spare parts and tools (but not including tires/wheels which are more). The FAQ for the FFR spec racers contains a list of the spares we carry (but does not include major items like tranny, etc.)
http://www.evisory.com/Racing/FFR_FAQ.doc
Hope that helps! Feel free to contact me and I will do what I can to help you get started!