My first cat was rescued, but not from a shelter. My second cat came from the local humane society (she was 3 years old).
She's a sweetheart and loves tummy rubs and scritches.
At that humane society, the adult cats are spayed/neutered and given their shots.
For kittens, if they are not old enough, you're expected to bring them back for the procedure and I think they give you a voucher for some amount off the cost of the surgery.
I'm not sure about the dogs, I've never gotten one from a shelter. First two dogs my family had were abandoned by their previous owners (one was living in the empty house they'd moved out of, yeah). The third dog was one someone couldn't keep.
They had me fill out some paperwork and went over it with me - things like expectations, if there were other pets, that kind of thing. They have you bring the cat to a vet within two weeks, and if you don't have a vet they will give you a very long list of vets they work with, some who provide the initial exam for free.