I think a much better idea would be that you join an existing nonprofit organization that promotes animal adoption, educates people about spaying and neutering their pets, etc. You could get started *right away* with helping animals, instead of spending many months getting paperwork and fundraising and financial record keeping together. Later, after you understand how such programs work, you could still start your own nonprofit if you think you could do an even better job on your own.
If you absolutely are bent on starting a nonprofit right away:
Before you file any paperwork, you need to get a very impassioned group of people together who support your idea and will provide some start up funding and serve on the board of directors, you need to get letters of endorsement from existing organizations saying this program is needed, you need to have data that shows your target group is in need whatever service it is that you want to provide and that they are not served otherwise, you need to have profiles of your staff showing they have experience and expertise, and you need to have a draft of what your first year's budget would look like (what expenses you expect to incur, what you expect to pay staff, etc.)
Also see:
Starting a Nonprofit Organization
http://www.managementhelp.org/strt_org/strt_np/strt_np.htm (one of my favorite resources)
Starting a Non-Profit
Checklist- The Things You Must Do to Start a Nonprofit Organization
http://www.ctnonprofits.org/pages/NonProfitResources/StartNonProfit.asp
Starting a Nonprofit Organization - How to Start a Nonprofit http://nonprofit.about.com/od/nonprofitbasics/ht/startingsteps.htm
How to Start a 501c3 Nonprofit Organization
http://www.wikihow.com/Start-a-501c3-Nonprofit-Organization