The best way to save up money for a horse is ... well simply to not spend any
just kidding it depends how old you are, I live in France so I don't know the rules in the States...
Actually, not spending money is a great way to save money.
Try putting yourself on a budget. Figure out how much money you have coming in - from odd jobs, gifts, regular income, wherever you are in the income scale - and tell every dollar where to go.
For example, decide on exactly how much money each month you plan to blow on coffee or movies or eating out - put that money in an envelope and only spend what's in there for those things. Run out of money in the envelope and stop doing those things till next month.
Make envelopes for regular groceries, clothes, gas, whatever your monthly expenses are (and if you are a kid, at least while your income is smaller, so are your necessary expenses). The tighter you can squeeze down your budget, the more money you have to go into your account set aside as your horse fund.
If you are old enough to have a credit card, don't use it. Paying cash gets you to spend less over all, so you can save more over all. Plus, cash is much more self limiting - when you run out of cash for any particular area, stop spending.
It may take a few tries to figure out what you really spend in certain areas, but once you decide what to spend ahead of time, you'll find extra ways to save.