I dont want to offend you but you need to make it through the beginner stage before you even consider rodeo. Speed events, while they are fun, are not for the light of heart. You need an above average seat along with fine tuned cues to get a barrel horse to respond appropriatly. Everything looks easy when a trained rider is doing it. Hell, I think I could give a real go at dressage...in reality, I dont have the skill set.
You need to find something calm cool and collected. This is where your journey starts. If you eventually want to ride rodeo, GREAT! Just dont get too far ahead of yourself. Start small, build from there.
*Because* the ad mentioned rodeo, I would scratch her off the list. I would be going for tank if I had to chose any. You only learn speed by going slow.