Yes, you have to decide whether it is to be a hobby or business. If it is a hobby, you are supposed to show all income, and can write off expenses to the extent of your income (I believe that is the difference). If it is a business, you can actually show a loss - for a while, at least. The idea is that you are really trying to make a profit, and eventually will either make a profit or get out of the business.
As a business, (and even as a money generating hobby) there are LOTS of deductions - anything that you need to do to earn a profit. That includes mileage to get to where ever you need to go - shows, FedEx, office supply stores, to customers or suppliers, anywhere that is required to do business. You can also claim show expenses such as food and motels, if you sell or buy at the shows, and it is part of your business. And, of course, any required permits, licenses, etc. It is true that some big equipment type purchases will have to be taken over many years. But a certain amount of even those costs can be taken ("expensed", I think they call it) in the first year - ask your accountant about that. You can also deduct not only the actual cost of the animals, food, and supplies, but also electricity, phone, and other needed expenses. However, if you don't have a separate building with its own electric meter, and a separate phone line, it can get tricky to partition personal vs business use. It can still be done, but it can trigger red flags for audit, so I suggest using an accountant. My CPA figures all this stuff out (and he makes WAY more $$ than I do - his is the business to be in for profits, lol!), so I only know how to do the bookkeeping to give to him, and he puts it in order and fills out the proper forms.
Although it is really difficult to be profitable in this business (especially in this down economy), there are real tax advantages to doing it as a part time side business, and trying to make a small profit so that you can deduct a lot of expenses that you might incur as a hobby anyway. Just talk to a good accountant to be sure you are on the right track, and not likely to trigger any unnecessary audits.