Sorry not trying to discourage you, but this is how it is. Also, if you want to shoot at a show, I would suggest you contact show management first and don't just show up and start shooting. I get very angry when I have put in a ton of time and money and I go to a show and see others poaching at the shows that I have booked with. It's not a good way to make friends with other photographers who could potential help you or teach you in the future. The horse world is small, the equine photographer world is smaller... If the show currently has a photographer, I would suggest you not try to sell your photos, because ultimately, you will be stealing sales from the photographer that is already there.
If there is no photographer and you feel up to it (is your work good enough yet to sell yourself as a photographer or are you just learning and aren't totally sure of yourself yet?). If it's the later, don't offer your services to the show just yet. If there is no photographer, go and practice for the day. If you happen to get some nice shots, perhaps go around and let some of the riders know that you go a nice shot. If there is NO official photographer at the show, it is usually ok to hand out business cards.
If there happens to be a photographer at the show, contact the show photographer and ask if you could shadow them. The answer will depend on the photographer. I know a lot of photographer who feel they ahve gotten burned doing this and will say no, but it still doesn't hurt to ask. If the photographer does say yes, whether they say it or not- but most will and may even require you to a contract, DO NOT post proofs online for sale. Instead, use the images for your portfolio and for the learning experience of it. Your portfolio is going to be what will help gain you more shows in the future. If your portfolio displays nicely composed and exposed photos, with the proper timing for strides and movement, you are more likely to recieve a yes from shows as far as you being the official photographer. If your portfolio is full of improperly exposed, poorly timed image, poorly composed images, you aren't going to get many shows that are willing to book you.
As far as the other questions. Typically riders and exhibitors will buy prints and it's on a per print basis. $30-$50 for an 8x10 is average depending on your area. Look up other equine photogs in your area and check out their pricing to get an idea of where your pricing should be. Be honest, if your photography isn't as good as some of the other photographers, it's ok to price yourself a little lower, but not too low. Likewise if your work is truly outstanding and different don't be afraid to price a little higher. Another popular thing right now are digital files and CD's. Many photogs are offering a price per file. If someone wants bulk files, sometimes they offer a bulk discount. Other photographers will offer a CD with all the images from the show (for the particular rider) for a set price (anywhere from $100-$350). I think it's becoming more popular, cause many people want to post things on facebook, myspace, etc.
Shows will not pay you to be there. There are many photographers that offer $5 prints, so why would they want to pay you, if they can get that for free? It used to be customary for shows to sign a contract with the offical photographer stating exactly what the photographer was offering to the show (free images for the shows website or marketing materials in exchange for exculsivity to photograph the particular show and links to the photographers website, as well as advertising that the photographer was on the grounds, etc. Also the contract states what is to happen if other photogs are there poaching- usually if caught, the show comittee will through the poaching photographer off the show grounds.) Now it very rare that you will get a show to sign a contract. In fact, many larger shows now will have multiple photographers. Some shows like (Dressage at Devon), actually will make photographers buy a photo pass. If you are seen photographing with out a photo pass, you will be asked to leave the show.
I have seen photographers try to do a prepaid thing for riders. Like if the rider wants to be photographed at the show, the rider will sign up, list their classes/ride times and prepay like $10-$20. That money that is prepaid, turns into a print credit which will be deducted from the order total, when they go to order their prints. If they don't order anything, it gets refunded or put towards a future show. I have tried that, and really didn't have much interest. Most people figured I was there, so i would photograph them anyways so there was no need to prepay. I do try to photograph everyone, but sometimes it is not possible. Still with parents and friends that have cameras, I really did not have a lot of interest in people prepaying. In fact many riders complain about it (check out message boards).
I'm not trying to discourage you, but show photography is not a get rich quick thing. I have had shows where I grossed $5k-$6k (not all at once, but over the course of a year). And I have had shows where I had to travel, stay in a hotel and figured I would make a decent amount and really didn't get much in the way of orders. Right now, it's hit or miss, and my order totals get lower every year (because of reasons previously discussed). And because of that, I have dropped shows that just werent worth my time. It's not fun, and I enjoyed when my schedule was booked, but when you are losing money it's not worth it.
So no, the short answer is you most likely are not going to make much money at this. My suggestion to you would be, if you have an in with a barn (that shows), offer to shoot exclusively for their barn (meaning don't shoot or sell to anyone at the show but members of that barn). Offer them a print or a CD for a set price, something that will work for you (again make sure the price is in line with the quality of your work- don't charge $200 for poor quality images). That way, you are making some money for your time, and still shooting, without the headache and responsibility of being the Official Show photographer. Also, in that case, I STILL would contact the show photographer-if there is one- and let them know that you will be shooting for that specific barn, but are NOT going to offer your services to other exhibitors at the show. In other words, make nice with the show photographer, and show that you respect them and are not out to steal sales. The day of the show, go introduce yourself to them, so that when you are shooting mistake you for a poacher and potential have you thrown out of the show. Plus, like I said, it's good to make friends with them.
I know personally, if other photographers give me the respect at a show, I will give it back. I don't having photographers come to where I am and shooting the same exact things I am. It really makes me angry. Make sure you give them their space and stuff. I am always happy to offer advice, to people that appraoch me and introduce themselves. There have been occassions where I have been contacted by shows and asked if I knew another shooter because the photographer for another ring couldn't make it. On those occasions who do you think I am going to ask? The photographer that was being sneaky and trying to steal sales from me, or the photographer that was polite and came up and introduced themselves to me and we created a repoire with each other outside of the show after? Just saying. :)