Andrea, I think it is wonderful that you want to help senior and fospice dogs - I have a soft spot for them as well and live with 5 senior Chihuahuas. Without trying to dissuade you from starting your own organization, I want to suggest that you might want to consider joining an existing group and band with them to raise funds, foster & adopt out this special population.
Starting a nonprofit isn't technically hard, I purchased a book online called starting a nonprofit for dummies and just following the steps. However, it is expensive, time consuming, requires a lot of annual state and federal filings and once you're in, it's like the mafia and you can't get out. I'm speaking from experience and wish someone would have shared this with me back in 2005 when I started my own nonprofit. Depending on what state you live in you will have to find a certain number of people to sit on your board and I've found that unless they have the same passion and drive as you they aren't going to give their all and uphold their duties (which is mostly fundraising and setting the goals of the organization and making sure those get followed through on). Board members also drop off and it can be difficult to find a replacement.
Also, unless you have a good network of dedicated volunteers or partners, the bulk of the work will fall on your shoulders which can be very overwhelming (keeping up with the database, all medical records, vet appointments, adoptions, returns, banking, paying bills, fundraising, filing annual taxes and reports). If you aren't already involved in a local rescue, I would definitely suggest that you find one and volunteer first. Learn as much as you can from all aspects of the organization and then decide if you want to take it all on yourself.
Regardless of what you end up doing, I am grateful there are people like you out there with compassionate hearts who want to help. Good luck!