General:
* Animal caretaker
* Animal handler
* Animal welfare worker
* Animal caregiver
Specific to certain animals:
* Veterinarian (for sick or injured animals)
* Veterinary technician (assists veterinarians)
* Zookeeper (for animals in zoos)
* Animal trainer (for animals used in entertainment or work)
* Farmer (for livestock)
* Pet sitter (for pets in people's homes)
* Dog walker (for dogs)
* Groomer (for animals' hygiene)
More informal:
* Animal lover
* Pet parent
The best word to use will depend on the specific situation.