Obviously there is A LOT when it comes to barrel racing, since people have dedicated their whole lives to learning about the sport and since entire DVD sets and books have been written on the topic.
My biggest advice: Take it slow.
Don't push your horse faster on the pattern than they are ready. Going too fast too soon will create bad habits, frustration (for both horse and rider), and can sour your horse on the pattern.
I would advise you to get some resources to get started. A trainer is the best resource, so keep looking for one. If you've got fairs that have barrel racing, there's bound to be someone around who trains. Also, any books or DVD's you can get will help with your foundation. Even if you are only going to do it for fun, it is good to have a general overall understanding of the sport and how you train for it. A few of my favorites are Charmayne James, Sherry Cervi, Connie Combs, Ed and Martha Wright, and Martha Josey. You can usually get used copies for a decent price on Ebay
. There also are some training videos on YouTube. Sherry Cervi has got some good ones; ExpertVillage has some okay ones.
You don't have to have western tack to do barrel racing, but of course it is nice to use the proper equipment for the sport you are doing (kind of like a basketball player wearing football shoes and pads .... you can do it, but it's much easier to just wear basketball shoes and your uniform). Check local tack shops for used name-brand saddles as they will hold up better than a cheapo off-brand saddle will. Good name brand saddles I like are Circle Y, Billy Cook, and Tex Tan.
Start your horse on the pattern with a smooth snaffle. If you can't do something (other things besides barrels too) in a snaffle, then you shouldn't be doing it until you train your horse better. A curb bit is meant to be a finishing bit and should never be used to "gain control" of your horse.
And in general your horse should be responsive to your legs and seat, direct rein and neck rein easily, stop when asked, and can be asked to travel at any gait and hold it. Your horse should be able to do all the fundamentals required in barrel racing before you ever show them a barrel --- make sure you horse is 100% broke!!