1) In high school you must display an interest in learning and the sciences in particular. Throughout your education, you should really work on your writing skills - communication is very important!!! Try to get a job or volunteer position working with animals, at a vet hospital, a local animal shelter, or animal rehabilitation facility.
2) Apply to colleges that are strong in biology and/or animal sciences. These two programs usually differ in focus - animal science usually focuses on applied aspects of animal production and care, while the biology track tends to focus on physiological, ecological and evolutionary science. Both tracks can allow a graduate to get a job in animal behavior right out of college. A third possibility is to apply to programs in veterinary technology.
3a) If you are interested in becoming a veterinarian, you apply for graduate programs during your senior year of college. There are very few vet schools, which makes the competition very tough. You must work very hard in college academically and display your drive and commitment by working or volunteering with animals in some way in order to have a good chance of getting in. Many students work at vet hospitals during college.
3b) If you are interested in research in animal behavior or physiology, you apply to graduate school during your senior year of college. Animal behavior graduate programs can be in either psychology or biology departments. There are masters programs that are typically 1-2 years or doctor of philosophy (phd) programs that are often 4-7 years long. To increase your chances of acceptance in these graduate programs, during college you should consider internship programs in an area of research that is interesting to you. There are many of these programs - check out animal behavior REU (research experience for undergraduates) programs. You can get a job with a masters degree in many different fields. If you go for a phd, you tend to really love research and school. Most people tend to stay working in universities. You can become a professor or a researcher. There is usually teaching involved even if research is your primary job. It is a long but very rewarding process!