I didn't start running my horse on barrels till he was about 5 and a half. I rather build a strong foundation on them first. Alot of the horses running in the pro rodeo are in their teens so like mentioned unless you want to do futurity runs, theres no rush. Do plenty of different things with your horse, go slow. Perfect your horse at the slower gaits before asking for speed. Mess around in warm up classes. It all depends on your horse for when to ask for speed.
IMO, a lot of the younger horses arent just mentally equipped to handle RUNNING the patterns and being hauled. Don't let your lack of a competitive horse cause you to push this one (not saying you are).