That's a really great idea! Founding an organization takes a lot of work, but it can be worth it--it's like having a pet in that respect, except it's even truer in this case. Here are 3 approaches--and you can definitely take more than one if you want to and have time:

It might be easier to get things done working with an existing organization, though. There must be at least one in your area, but you'll have to check and see if any let people your age join or volunteer; some don't unfortunately. I've had that problem before. Anyway, if they have any projects you can take part in, do those! You can also make a donation if you've extra money (this works for organizations outside your area as well); if you want to donate without actually spending anything yourself, there's a search engine called Goodsearch that lets you do that. You might have seen advertisements for it on television. Every time you use it to search the Internet, it donates a cent to a charity of your choice. It doesn't sound like much, but I know that the ASPCA, for one, makes quite a lot of money that way, and it does the things you mentioned. I personally use that website although the organization I use is about wild animals, not pets, and I've worked with Girl Scouts and AABY (an organization that does the thing you mentioned, but only for dogs) on projects helping dogs. I think we did something at their thrift shop and made dog biscuits and things like that. Oh, and you may be able to walk dogs and such at shelters in your area to give them exercise and such.

If you want to start your own organization, you can read The Teen's Guide To Global Action for advice that's much better than any I could give. Well, really, it's good for doing any sort of community service, not just starting your own organization, but that's hardest by far, so it's what you'd need it for the most. I recommend making a website for your organization for publicity (It's best if it has its own domain name, but that costs money and is unnecessary), but you don't have to, of coursde. You do have to tell people about it somehow, though. Are you allowed to put up posters around town where you live? That could work. Also, you could just tell people. And if something happens where you live that's disastrous for them, you can write to the editor of your local newspaper about it. I don't really know; I haven't started an organization before, but people our age and younger have succeeded in it before, and they must do it somehow.

If you find a stray animal, it's best to take it to some place like an SPCA that's known for treating them very well and not killing them, if it's at all possible--or keep it, of course, but that's not always possible. And of course neutering your dog is a good idea if you haven't already, but I'm sure you've heard that.

Good luck! Sorry that was so long. Also, I'm sorry I didn't post here when I was being called; I just moved, so I lacked Internet.