On average, how much does it cost to get started in this sport? (Say, for tack, riding apparel, etc.)
Are there any disciplines in the horse world that aren't expensive?
It's expensive. I agree with what people have said, you get what you pay for! But it can be done on a budget if you're careful and look around. I'm all about ebay
and tacktrader for quality used tack. You don't need the best of the best but you do want something that's going to last and fit you and your horse well. Can I train my own horse? He's a very tall quarter/fox trotter mix and very smart. I think he'd enjoy learning to do this. If so, what are some tips for training him to jump?
If you've never jumped before I'd advise against it. Jumping is hard and you need to be secure and stable over fences before you start teaching your horse how to do it. It's easy to teach your horse bad habits while jumping and can be dangerous if you don't know what you're doing. If you're dead set on training your horse yourself I'd sign up for a bunch of jumping lessons with a quality instructor. Then have your instructor work with you to train your horse when you're ready. The best way to prepare a horse is good flatwork
Do I need to put the horse on joint supplements? If so, what are some good brands to use?
Npt necessarily. There are those who swear by putting their horses on supplements "just in case". There are those who swear by only using them when necessary. Good arguments for both sides. I'd talk to your trainer and the vet and see what they say. Is there anything else I need to learn about show jumping?
Get ready for some FUN.