If you grew up with a family pet or have spent time on a farm, you're probably already very familiar with how to care for animals. But if you want to gain experience handling a large group or different kinds of animals, contact your local zoo about volunteer or part-time positions. Many zoos have programs in place to introduce young people to the duties and responsibilities of an animal handler. If your local zoo doesn't have such a program, try to create your own. Contact zookeepers, express your interest in their work, and ask to "shadow" them for a few days.
Many part-time jobs are available to high school students interested in working with animals. Pet shops, petting zoos, stables, and kennels are likely to have a few after-school positions. In larger cities, you may be able to start your own animal care business as a dog walker or pet sitter. Or look under "animal handler" in a local business directory. Some animal handlers work exclusively with movie production crews and other entertainment venues; you may be able to work as an intern or temporary assistant on a production.