If you don't have money, but you have some free time, there are literally thousands of shelters and rescue organizations across the world that would love for you to volunteer some time with them.
Many of these organizations are government or donation funded, which means their resources are exceptionally low. Any time you can give them will make a difference. All of them will provide you with the required training, and all of them need your help. A good place to start is by checking the VolunteerMatch website for opportunities in your area that work within the amount of time you have to offer.
Additionally, if you don't have a lot of free time to contribute, there are still things you can do. Most shelters have a "wish list" of items they frequently need during the course of the rescue operations. Items such as pet food, pet carriers, blankets, etc will be gladly accepted as a donation to your local animal shelters.
Some organizations will even be able to work within your schedule! If you can help them write articles, update their website, send emails or even stuff envelopes - many of these jobs can be done in your own home on your schedule.
The first step is to find an organization you wish to support - then find out what you can do to help. If the organization does not have a specific volunteer form, be sure to list your skills and available time. This will help them find an opportunity that is the best fit for you.