I am thinking about volunteering at an animal shelter, but don't know much about them.
I think you should working at an animal shelter or pound is very fulfilling 
Are there all kinds of animals there or just dogs?
 Mostly dogs and cats, and sometimes sick or wounded wildlife. Though my shelter has birds as well.
Have animal shelters been around for a long time?
One of the earliest comprehensive measures was the "Georgia Animal Protection Act" of 1886. so they have been around a very long time 
Don't some shelters put animals to sleep if they can't find a new owner. I don't think I want to volunteer at one of those
Animal shelter
An animal shelter or pound is a place where stray, lost, abandoned or surrendered animals, mostly dogs and cats, and sometimes sick or wounded wildlife are kept and rehabilitated. While no-kill shelters exist, it is sometimes policy to euthanize sick animals, and any animal that is not claimed quickly enough by a previous or new owner. In Europe, of 30 countries included in a survey, all but four (Czech Republic, Germany, Greece, and Italy) permitted the killing of healthy stray
of dogs and cats. She has starred in a one-minute video PSA, urging everyone to adopt animals and have them sterilised. She has been quoted saying: "Breeders, pet stores and people who don't sterilize their animals are responsible for all the animals who end up homeless. Every time someone buys a dog or cat from a breeder or pet store, a homeless animal roaming the streets or waiting in an animal shelter loses a chance at a home and a good life." She also promotes breaking the