Good question, but it’s difficult to give a specific answer, as I think this really depends on what you’d like to do – some jobs that involve working with animals might involve some more specific training – for example, if you’d like to be a vet, you would need a veterinary degree to do this, as you would need to learn about the clinical side of looking after animals.
If you wanted to work in an aquarium, or maybe at a safari park, you might need different qualifications that are more focused on animal husbandry (i.e. looking after their health and welfare) and animal handling, nutrition, etc.
I’ve found it can be really useful to write to places like this and ask if you can come along for a day, or a week or two, and shadow some of the staff who work there. It’s a great opportunity to learn more about something you’re interested in, and it gives you the chance to ask specific questions of people who are doing something you might want to do as a career.
When I was at school, I wanted to be a vet, so I chose science-based A-levels, and then went to university to study veterinary medicine. Since then, I have moved on to work with aquatic animals instead, but my degree is still useful for that.