Any breed will be able to do recreational riding, and as long as it's quick and can turn nicely, it can be taught to do barrels.
I would just keep an open mind as far as breeds, and when you find one that clicks well with you, then snatch it up! I've ridden everything from ponies (various breeds), to Belgian drafts, and each horse is talented in different ways, but all of them can do recreational riding. For showing, I've gamed Arabs, QH, TB, Saddlebreds, and other grades, and all of them can be competitive.
"The ideal horseman has the courage of a lion, the patience of a saint, and the hands of a woman..."