Where to start.......
Obviously, many professionals have dedicated their lives to understanding and perfecting the art of barrel racing, so there is way more to know about it than you can learn on any chat forum -- but it's a start.
First I would recommend for you to start reading up! There are lots of great books and DVDs out there that can really give you a foundation on all aspects of barrel racing such as training, tack, picking out a prospect or trained horse to buy, the dynamics of the pattern, health and horse care, traveling, and much more. A few of my favorites are Charmayne James, Sherry Cervi, and Sharon Camarillo, but there are LOTS of resources out there.
You can get the books or DVDs used for cheap prices on either Amazon or Ebay so I would check that out.
Also, since you don't have any experience with barrels, it would be best for you to find someone who IS who can help you select a barrel racing horse. Since you are just starting, I would recommend looking for an older finished barrel horse that has been there and done that. You will learn the best by having a horse that already knows what they are doing.
Yes, typically quarter horses are the best for barrel racing but there are always exceptions to the rule. There have been many paints, thoroughbreds, and appendix quarter horses (and more breeds) that have been very successful at barrel racing. Yes, the horse has to be fast, but more important is a good mind and a big heart. A horse that TRIES 100% everytime cannot be given a price.
The best advice I can give you is to START SLOW. Yes, you may purchase a horse that can go fast because they know what they are doing but you do not want to develop bad rider habits that can happen if you start going too fast too soon than what you are ready for. Barrel racing is much more than just hanging on while you whip and spur away at the horse. Your body position and your cues are SO important to guide your horse through the best pattern possible. No one will judge you for taking your time and going slow because everyone has to start somewhere.