Bulls Head falls under New York City's Community School District 31, which covers all of Staten Island, and its schools are a real part of the neighborhood's family-first appeal.
The zoned elementary school is P.S. 60 Alice Austen on Merrill Avenue, with I.S. 72 Rocco Laurie as the local middle school. Most of the neighborhood is zoned for Port Richmond High School, and the College of Staten Island High School for International Studies is another well-regarded option in the area. Just north, the Staten Island School of Civic Leadership serves nearby Graniteville.
Catholic and private options are dense here too, including Moore Catholic High School and Bishop Patrick V. Ahern High School, both within Bulls Head, alongside the College of Staten Island right on the neighborhood's eastern edge.
I don't grade schools for you, that's your call and your family's. What I'll do is point you to the zoned schools for any specific address and the current third-party ratings on Niche, GreatSchools, and U.S. News, so you're deciding on real data.