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
You're right no-kill shelters exist, it is sometimes policy to euthanize sick animals but its only when the animal is suffering and in pain.
I have a big decision to make. Any other advice to help me decide?
Well critics believe the new term "animal shelter" is generally better than "pound" so I suggest you only use the new term from now.