Organized animal welfare didn't begin until the 1820's in Britain, when a number of philanthropists founded the society for the prevention of cruelty to animals. What you have to remember is, at that time animal welfare was considered a waste of time and money, and animals were not recognized as sentient beings by most people. However with a lot of education and hard work the first SPCA gained enough momentum that by 1840, Queen Victoria granted permission to rename the organization the royal society for the prevention of cruelty to animals.
Animal rescue found its way to the US in the 1860's, when a man named Henry Bergh founded the American society for the prevention of cruelty to animals, or ASPCA. Other rescues soon followed and within the next 3 years there were anticruelty laws and rescues in at least six different states. National organizations soon formed with the American Humane Association (AHA) in the 1870's, and the Humane Society of the United States in 1954. During most of this time the organizations were more about educating the public and had not formed actual shelters.