The State University of New York is a multi-campus university system that's publicly funded. It's larger than the University of California, which is another, MUCH MORE prominent public system of higher education.
The four SUNY medical schools (Buffalo, Stony Brook, Downstate (Brooklyn), and Upstate (Syracuse)) generally take in students who are New York State residents, have outstanding undergraduate careers, but the SUNY medical schools show no preference for graduates of SUNY campuses.
If you're thinking of a SUNY school for college, I'd suggest SUNY Binghamton which has an outstanding national reputation and is considered the most selective of the SUNY campuses. For what it's worth SUNY Buffalo is the university's flagship campus (Berkeley is UC's flagship, Ann Arbor is UMich's flagship), but its undergraduate program and its students are a little weak compared to Binghamton.
Good luck.
Tim of New York City.