Use online sign language resources such as the official American Sign Language website or the Signing Savvy website to learn and practice sign language. These and other sites have a number of free resources available, including sign language videos, dictionaries and helpful learning games and tools. Some sites require you to pay after using a certain amount of the sites' resources, but if you take advantage of the many different online resources, you should be able to make significant progress in your quest to learn sign language without paying anything.
There are a number of books on sign language that can be checked out from your local library. Libraries are an often overlooked resource in the digital age, but for learning sign language, there are few better options. Books are a great medium for displaying images in an organized fashion, which is what learning sign language entails. When your sign language book is due back at the library, simply renew it. If someone else has a hold on the book, you may have to return it and wait several weeks before getting it back.
The best way to learn sign language is probably to get out there and practice with other sign language users. If you don't know someone who uses sign language, you can usually volunteer at a local deaf school or center for the hearing impaired. In return for your services, you should be able to practice your sign language with the community for free.
Besides the library, there are other places you might be able to get ahold of sign language learning materials for free. If you know people who have learned sign language, most likely they held onto their class books and worksheets. You can ask to borrow these to help you learn. You can also visit local schools to see if they have any unused sign language learning materials.