What is the word to describe bringing up your pet?
- Raise sounds odd.
- Breed sounds weird.
- Rear seems to be more for farm animals.
Breed seems inappropriate because OP refers to pet in the singular. If you breed [some type of animal], this implies producing significant numbers, often on a commercial basis and/or with the intention of improving the genetic stock over generations.
I don't think that keeping a dog/cat, for example, necessarily implies you've had the animal since it was a puppy/kitten.
Google Books says "reared kittens" is three times more common than "raised kittens". But scanning the results, my impression is that in the vast majority of cases, rear is used in clinical/research contexts (lab-bred kittens reared in the dark, without social/maternal contact, etc., which aren't connotations OP would want). Many of the rest involve hand-reared - again, a specialised context implying weaning without the aid of the natural mother.