Or any animal based non-profit, for that matter. As a founder of a non-profit horse rescue that started over ten years ago, I'm often asked advice in getting started. Here are some general tips I tend to give.
I also am a founder of a 501(c)3 rescue and also advise you to volunteer at the type of rescue you are interested in first, as well as maybe consider fostering for a group so you can get experience with the care, adoption process, etc., as well as reporting to a board and following policies.
Do a ton of research. For example: I see so many brand new groups pop up with improperly formed boards (you can't have a majority that are related, for example, I often see boards of 3 people and two are either married or family), or don't run it as a business right away.
And the goal should be self-sufficiency as soon as possible which means only helping horses that you have funding for. We literally started by taking in one horse privately at a time because that's what we could afford, but as it grew and we incorporated we set a one year goal to be completely self-supporting. That meant maybe only 4-6 horses at a time in the rescue while we built our reputation and support base. We now average between 40-60 horses at any given time and are at a point we are discussing the need for staffing which may mean reducing the amount of horses short term if we can't increase funding enough to cover it.
In that regard, there's so much competition for the major grants, so focus locally. Get to know your local law enforcement. Form partnerships with businesses. Look into community foundations and local businesses that might have corporate giving and grants.
Be sure and setup any property arrangement legally as well. I own the property the rescue is on but have a legal lease spelling out the use of the property and division of things like repairs - who is responsible for what. I maintain a separate liability policy on my homeowners for my personal horses, and the rescue has liability insurance separate. Don't forget about insurance!
And the personal horses ... make sure you account for their expenses. You need to be able to show at any given time that you are covering your personal horses' expenses in one way or another; either buying things separate, or paying the rescue board or fair share of expenses, or tracking work time to "work off" board - be sure and document it. Maintain separate accounts at all vendors - vet, feed shops, etc.