Check your local newspaper classifieds or Facebook sale/trade groups for unplanned kittens or puppies free to good homes. Additionally, while animal shelters do require an adoption fee, that fee is simply enough to cover whatever medical care (vaccines, spay/neuter, etc) and food each animal would have received in their care.
Both options are an excellent resources for finding animals of all ages and giving them a second chance at life in a loving home. While you may not need to pay any money up front for a pet obtained via classified ads or a local sale/trade group, it is very important to take any new pets to a veterinarian the first chance you get. Check-ups, vaccinations and spaying/neutering are very important, as they keep pets healthy and prevent any more overcrowding of animal population.
In the long run, it may end up being a less costly option to adopt an animal from a shelter that has already given the animal its initial vaccinations and medical exams. Many shelters also spay and neuter animals. Many shelters also have adoption events that offer animals up for adoption with a lower adoption fee than they would normally charge. Check your local animal shelters for event schedules.