Start saving! Get all the essentials listed with price ranges and considers how much it will cost. Then get a professional to go with you like a trainer. Once you find a horse you like ask if you can test ride it and if you like it get a vet to go out and check it they can tell you if there's anything you need to worry about.
Best advice would be to get a job and start saving! I recently bought my horse all on my own! After getting a job, tell yourself you're going to set aside a certain amount of money each paycheck (for me it was $100 each paycheck, for you it may be more or less!). Tell yourself you won't tap into that money after it's put away unless it's an absolute emergency (this part is huge as you will want to spend the money on other things!) This will take a while, especially if the horse you're looking for is expensive, but it'll be worth it! Good luck!