Most people who live near school stay at either Kips Bay Court, Stuyvesant Town or Waterside Plaza. People will typically convert the living room into a bedroom and fit one more person in there. I did that both last year and this year and my rent has been around 1250 a month for my own room.
Res housing is actually really expensive and, though close, proves to be inconvenient. The res housing follows the undergrad schedule (closes on May 15), but dental school ends May 30... in short, you get kicked out two weeks before finals. If you want to stay through finals, they charge you an extra fee that is $1000+...
Like Musl said, there's definitely a trade off in that living outside of manhattan will allow you to save money, but will cut down a bit on study time and social participation. If you're moving to NY to experience the city, you may want to consider staying in Manhattan.
As for the OPs question... it's really up to you... living close to school is really convenient... but coming home to a warm meal would be convenient too. Also, consider how well you're able to study at home... are your parents the type to keep distracting you?
Utilities -- don't really cost that much, as most apartments have them included. In places where they're not included, I'd say maybe 30 a month for utilities and 20-30 a month for cable internet and possibly tv depending on what package you get.
Food varies a lot... I personally spend about 600 (usually closer to this) to 1000 (when too many friends visit) a month on food, but I'm a foodie and I really enjoy eating out and trying new places... and NY just has A LOT of good food. If you stay home and cook more, or opt for cheaper options when eating out, I'd imagine you could get by on less...
