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.
I agree that you should find the individual that best suits your needs and not worry about sticking with a specific breed--
However it would just be WRONG if I didn't recommend an Appaloosa-- Appaloosas come in a variety of types and colors, and there are some excellent games and trail type Appaloosas being bred, raised and trained and shown in Arkansas and surrounding states.
This guy is not a bargain-basement guy, as he has World and National top 5 and top 10 placings, and was the overall year end highpoint games horse in the state of Arkansas two years in a row-- not sure if he is in your price range (he wouldn't be in mine at the present time, much as I would love to ride him!) but I bet his seller would not mind talking to you about trainers/friends/etc in the area she might know that have other Appaloosas for sale since she would definitely have a network of people she has contested and ridden with.
Also there are a couple of active Appaloosa clubs in Arkansas that hold shows, and most Appaloosa shows have a full slate of games classes as well as a wide variety of other stuff, and Appaloosa people are notoriously friendly and fun, so you might have some good opprortunities to go out and enjoy your Appaloosa at breed shows and meet folks and have fun, as well as open shows and trail riding and etc.
It depends on the horse themselves, the rider, experiance, ect. But I reccomend a QH. I love them, if I didn't own my wonderful OTTB (Don't reccomend one as a first horse, but anyway) then I'd have a QH. Although I most likely will stay with OTTB's/TB's from now on.