Congrats on your great ACT and SAT scores! Doing well on these tests will open a lot of doors for your future career plans!
Having a degree in engineering makes you very marketable once you are finished with undergrad, since a student who does well in an engineering program tells future employers that this student can work hard, learn new concepts fast and apply these concepts and skills to a variety of problems to solve. While at Duke for the past 4 years working on my PhD in biomedical engineering, I've met several biomedical engineering undergrads that take all the pre-med classes and apply to medical school. In fact, all of the undergrads (~4) that have worked in my lab have taken this route and all of them have been accepted to medical schools.
Some schools with very good biomedical engineering programs include Duke, Georgia Tech, Johns Hopkins and the University of California San Diego.
I would also suggest looking into mechanical engineering, electrical engineering and chemical engineering undergraduate programs for your undergrad since many employers are familiar with these classical engineering degrees, instead of biomedical engineering (which is a new field and is sometimes questioned as being a true engineering discipline, depending on who you talk to). You can major in a classical engineering program, and minor in biomedical engineering.
Additionally, when you are finished with your undergrad degree, you can take the tests to become a professional engineer (important if you want to go into industry, opposed to academics). As far as I know, in the USA, there is no biomedical engineering test, but rather mechanical engineering, chemical engineering etc, so you might as well get classical engineering training in undergrad to write the professional engineer test once you are finished with your undergrad degree. Just a suggestion. These tests I mentioned are USA specific.
I am not sure about what scholarships are available for you; however, you might want to look at the National Science Foundation website, the Howard Hughes Medical Research website and the National Institutes of Health website about national scholarships.