Have you ever volunteered for a rescue group (dog, cat, rabbit, anything)? I would suggest that you start with that so you will have some local rescue resources.
I had a small rabbit fostering group at one time – we applied for and got our 501c non-profit status from the IRS.
What I first did was help at some local rabbit rescues and I realize you don’t have that in San Antonio. However I also volunteered at a local no-kill shelter (dogs and cats only). And that helped me see what it took to start my own rabbit foster group.
I also was an educator and foster for the House Rabbit Society. I would recommend you apply as a volunteer with them as well – you will get access to their resources and you can also tell shelters and other rescues you are a volunteer with them.
I paid all my expenses out of my own pocket when I had a little foster group with 2 other friends. Fundraising is always the BIGGEST challenge of any of the rescue groups I had or helped with. I am currently a board member for a Guinea Pig rescue and run their rescue store and we are always in the red. In fact, any of the rescue groups I helped where always in the red.
As for getting a 501c – you first need to incorporate in the state of Texas and you will need a board of directors (at least 3 people) to incorporate. Once you are incorporated you can then apply for your 501c3 if you decide to go that route.