Your idea is very possible, but also very difficult emotionally, financially, and physically.
I started my 501(c)3 non-profit animal rescue in March 2008 when I was 16 yrs old. My mom helped with the driving and phone calls, but the rest was all my responsibility (i.e. The planning, fundraising, pulling animals, applications, contracts, medical and financial decisions, housecheck approvals, website, etc.).
There is also a lot of drama in the rescue world. Be prepared to find other rescuers who love your work, and others who are so very negative they become a hinderance to your work and passion. I am not talking just not agreeing with certain things, but other rescuers trying to screw you over and stab you in the back. In short, stay out of the drama and stick with those who share similar goals, philosophy, and standards.
To be reputable, you must require fosters and adopters to fill out applications, and your applications must be thorough and information grasping. You will also need contracts for all fosters and adopters. These not only legally protect your rescue and the foster or adopter, but they help protect the animal. Housechecks are vital; check the home for dangers and check the yard for dangers, including low points in the fence and holes in the fence. Always deny an applicant that does not allow a housecheck.
Always charge and adoption fee (adoption donation) for the well-being of the animal and to minimally help with future costs. In order to assist with the pet over population, all cats and dogs must be spayed/neutered. Being utd on shots and microchipped (back to your rescue) are strongly recommended.
Our rescue REQUIRES the animal to be returned to our rescue if the home can no longer care for him/her. This prevents the animal from going back to the shelter, back on the streets, or to a bad home.
Also, when naming your organization, keep in mind there is a difference in "rescues" and "shelters". Rescues are usually 501(c)3 non-profit. They have a higher chance of being no-kill. Shelters refer more to the city/county opperated facilities that are required to take in any pet that walks in the door, they are funded by the city/county, they usually DON'T have a no-kill policy, etc. They are typically very high-kill due to lack of space.
Are you going to have a facility other than keeping animals in your home and in foster homes? If so, call it a "rescue facility" and not a "shelter".
If you post some key terms you want to focus on with your rescue, etc., I can help brainstorm some names.
Lastly, if you are no-kill, be prepared to spend thousands of dollars on medical for certain animals, both scheduled and emergency. Some animals will go through your rescue requiring minimal medical attention, while others will require extensive medical attention (i.e. Orthopedic surgery, cardo consults or surgery, dental work, other surgeries, emergency vet care at a 24/7 critical care facility, and much more). No-kill means you do NOT euthanize for medical conditions that CAN be treated (even if you lack the funding). You can not euth due to lack of space. It also means you do not euthanize behavioral cases unless it is a very severe case and professional assistance is not working. In this case, euthanizing the animal is to protect the community, other animals, and the animal in consideration. We have rescued dog-aggressive, cat-aggressive, and slight human-aggressive dogs. They all went through professional training (approx. $900-$2,100 a dog) prior to adoption, and their adopter received free lessons with the trainer (at our rescue's expense).
In example of euth'ing to protect the community, if you have a dog that is dog aggressive and has killed another dog, there is no possible way you can place that dog in a home while being 100% certain you are not causing danger to other dogs. Eventhough the home may not have other dogs, if he/she ever were to get loose or a loose dog run up to him/her while on a walk, that would be a disaster waiting to happen.
Look for sponsors, grants, and donors. Hold fundraisers and be smart with your money, as there is never enough of it. Ask for item donations (beds, collars, crates, treats, food, etc.) instead of going out and buying them. Save your money for medical bills and food if not received in item donations. Also, for example, we usually have more animals in our rescue, but right now we only have 4. ALWAYS keep at least $2,000 in the bank for medical emergencies. While $2,000 may not even cover one ER visit, it is definitely smart to stay on top of your finances and plan for the "just in case".
We just rescued a stray sick 2 week old kitten that needed to be rushed to our regular vet, then we were transfered to the ER (as it was late at night and our vet was closing). Unfortunately, the kitten did not make it, but just in those few hours, we spent probably $500-- though that's the least expensive bill we have received from the ER... just so you have an idea). Those bills and bills from specialists usually run in the thousands ($1,000-$4,000 that we have had experience with). Also, try to hook up with vets that will work with you and give you a rescue discount.
I will post again if I think of anything else, but let me know if you have any questions regarding anything. We had our 501(c)3 status done by a friend that was a CPA, though it was still about $800 or so.
Rescue work is definitely a wonderful experience and it is a wonderful feeling when you can make a difference. While you know you can not save them all, the harder part is loosing an animal in your rescue from medical conditions. We only had that kitten for 5 days, but it takes a toll on you and the time spent with the animal is irrelevant when it comes to saying good-bye.
Here is our website you can check out if you would like: Our Mission - LEASH Animal Rescue