What courses do you recommend in high school or college to become a successful biomedical engineer?

In high school, I think that taking challenging courses in math and science is helpful before you go to college. Having a solid foundation in math and sciences made me confident in my base skills that I needed when I started to take Engineering courses. My high school was limited in the math courses that I took, but even my exposure to some Calculus in high school, I believe helped to make my understanding of Calculus better in college than if I had not been exposed to it in high school. My college did not have any prerequisite requirements, but I did take Calculus and Chemistry courses as well as the required courses that I needed for a Biomedical Engineering degree. I love the life sciences, that is one reason why I decided to get a Biomedical Engineering degree, rather than another type of Engineering degree.

Depending on which college you attend, there are specific Biomedical Engineering Majors. There are not many of these yet though the number is growing. Those majors are mixtures of being an electrical engineer, mechanical engineer, and biologist.
In the absence of an actual major, you should start by picking a major at your college that most closely resembles this kind of mix. perhaps an EE major, or a Mech E major. This will mean at a very minimum, you should definitely look at physics 1 & 2, biology 1 & 2, calculus 1 & 2, and possibly even chemistry 1 & 2. (Some schools will name them slightly differently but the freshman / sophomore level groupings from the core science groupings.)
At the high school level, if your school makes these kinds of courses available start now in the same kind of classes. If your school offers CAD classes try to get into one of those as well. This will give you valuable computer drafting and design experience that will serve you well in later years.