Cowboy action shooting (CAS) can be expensive to get started in, if you buy all the guns and gear up front. The good news is that CAS is generally a friendly and people love to share. At every club I shoot at, all you need to do is show up and look interested, and several cowboys will be competing for the right to loan guns to you.
Go to a match near you, and show up early. Introduce yourself as a newcomer looking to learn about this game. At my clubs, the only requirements for newcomers are that you know the basics of safe firearms handling, have a positive attitude, and are wearing long pants (no shorts allowed). We've got extra safety glasses and ear plugs if you didn't bring them. We'll find a holster rig that fits you, and set you up with borrowed revolvers, rifle and shotgun.
You'll impress folks if you bring your own eye and ear protection, and make a little effort to dress the part, by which I mean blue jeans and cowboy boots, if you have them. A cowboy hat would help you fit right in, too, but isn't required for first time shooters. I would also suggest bringing a few boxes of 12 gauge shotshells (must use #6 or smaller shot -- I recommend Winchester AA light target or Remington STS light target loads), and perhaps a couple of boxes of .45 Colt and/or .38 special "cowboy" loads. That way, you're not borrowing ammo as well as guns.
Doing it this way, you'll get to meet a bunch of cowboys and try out the game before sinking any real money into it. You'll also get a chance to see and perhaps try several different models of the various cowboy guns available, which will help inform your purchasing decisions. Once you've borrowed for a match or two, and you still want to play, you're in a good position to start spending your money wisely.
Once you do decide to buy guns, I strongly recommend starting with the revolvers. That's because it's a lot easier to borrow long guns than revolvers, due to the need for you to have holsters that fit the particular revolver model and barrel length. You'll probably drop $1,000 or so on a pair of revolvers and a leather rig for them, but it will fit you and your guns perfectly, and all you'll need to do is borrow someone else's rifle and/or shotgun at a match. You can spend less by buying used.
The "hot" CAS revolvers these days are the Ruger "New" Vaqueros that came out last year. The older model Vaqueros are still wildly popular, followed by the various Italian Colt clones. By going to a few matches and borrowing, you'll get a sense for what you want to put your money into.
Welcome and good luck, pard!