I live in one of the areas you mentioned (I've heard it called both Bulls Head and Graniteville)
School: The elementary schools in the area are good. In fact P.S.22 (on Forest Avenue) is well-known for having a good chorus (in fact, they went to Hollywood or something, so they're
really well-known).
Further south, there is P.S.60, which is also a very good school. It isn't well-known or anything, but it's been mentioned (positively) in the newspaper a few times.
For middle school, the zoned school is I.S. 72 if you live south of the Staten Island Expressway (SIE). If you live north of the SIE, you're child will be zoned for I.S.51, which has a bad reputation (I mean, some people overhype it, but it isn't known as one of the better schools. I.S. 72 is about average, and I.S.51 is below average for Staten Island)
Also, if your child goes to I.S. 72, they put some emphasis on learning languages. They offer Spanish, Italian, and French (though they might get rid of French), and they don't put that much emphasis at I.S.51)
The high school is Port Richmond High School (though your child can technically go to schools all over NYC). They put too much pressure on the students, in my opinion (as a student there). They feel that a lot of kids will drop out if they don't keep up the pressure (I mean, they aren't completely wrong, but they're blowing it out of proportion).
There is also a school near the SI Mall, but I don't know much about it.
For shopping, there are some small stores (pharmacies, delis, and things like that) along Richmond Avenue, and there are a few shopping plazas (one by Lander Avenue, and the other by Victory Blvd). You can also go up to Forest Avenue for some more shopping choices.
The predominant way of getting to Manhattan is by express bus. The X17 gets to Lower Manhattan in 30 minutes, and Midtown in about 60 minutes. During rush hour, the Midtown buses run through NJ and get you there in about 45 minutes.
The fare is $5.50 per trip, or $50 for a weekly pass.
The X17 runs weekdays and Saturdays. However, there is another route (the X10) that runs 7 days per week.
See the SI bus map:
http://mta.info/nyct/maps/bussi.pdf