Introducing a new cat

We have one cat and one dog, and we may be getting another cat. My grandmother has a colony of smei feral cats. Sometimes people abandon cats there and we fell in love with one of them. She is sweet and loves attention. Our cat has never been around another cat so im not sure how to go about introducing them.

We have a month since they both need fixed and we don't want kittens. The cat at my grandma's will get spayed first and my cat will get fixed in January. Should I wait until they are both fixed? Or bring home the female once she is spayed?