I catch ride at my local shows, and you need to make sure you are COMPLETELY ready. Its a difficult thing to do, as you may only have a couple minutes and one or two jumps with the horse before you show. Some horses may be great, others could act like complete brats. And if the class doesn't go well, the owner may blame it on you.
You want to ride as many horses as you can to prepare yourself, and to get your name out there try putting an ad up in your local tack shop saying you can school or exercise horses for people. If you do a good job doing it for free, you may get asked to show the horse, or they may offer to pay you. I schooled and showed horses for free for 3 years before someone offered to pay me. And even now I still school horses for free most of the time, and an owner may pay me $20 to show their horse over 2 jump courses and a flat class. It takes time to get up there, but if you eventually want to be a trainer, then its great experience.