Well here are the normal distances between each barrel around here....
90 feet between barrel 1 and 2.
105 feet between barrel 1 and 3 and between 2 and 3.
60 feet from barrels 1 and 2 to score line.
You should have a fairly bigger arena, especially to give you and your horse enough to learn without being crammed in a small area. You just need the three barrels. A normal sized one you can find just about anywhere...
As for you and your horse..a normal tack set would be a western saddle (barrel saddle), bridle, tiedown, brestplate, and some good support boots for the horses legs. I just use the front ones, but back ones wouldnt hurt. Make sure you have a good bit for turning and not just a tombthumb or regular curb bit. Try a combination bit with the rope or even just a plane old wonder bit..some people use hackamores or you can just start in a snaffle depending on how your horse is.
My daughter Tricia Aldridge teaches barrel racing and you can look at her website info at Fort Worth Stockyards Stables - Show Coaching she works in downtown Fort worth across the street from the cowtown coliseum. I do not know where you live, but you will save a lot of money and time if you can work with a trainer. Even if it is just now and then. My daughter made the fastest progress and went from playdays to winning a lot of shows after she started working with a trainer when she was younger.