Just to let you know, a horse suitable for quiet trail rides may not be suitable for barrels. Also, most riders use a special barrel racing saddle for the games, although in MY opinion, you don't need one. I'm afraid I have a bit of disdain for most young barrel racers having dealt with both the girls and with the horses they ruined the training on.
I understand there's probably tons of riders that are great, but I just wasn't dealing with any of them.
One thing you might want to keep in mind is that you want a "easy keeper", or a horse tha has good tough feet, doesn't need massive amounts of feed as opposed to mostly grass/hay and a quiet well trained animal. Too many people go shopping and fall in love with some high spirited delicate disposition untrained animal. Then it turns into misery as they try to work with an animal that is unsuited for what they want.
Horses don't automatically love people and want to prance about prettily and behave properly. Ex-racing thorughbreds, Arabians and some appaloosas can be high strung and hard to handle. Not that ALL of them are, I myself owned appaloosas, and had several ex-racers and arabians I retrained just fine. But it's just so much easier and better to look for the animal you want, rather than ending up with an unsuitable animal because you feel you're "rescueing" it or that it will behave just because you love it.
Horses can be really rewarding to own and work with. But without plenty of foreknowledge, they can easily turn into a nightmare of money and time and fuss. I'm really glad that you are learning all about the issues and horses before you jump right in. That's pretty rare, so kudos to you.