Narrow, squinty eyes

What things can we see and do to get a feel for NYC?
What things--museum, oddities, historical, cultural-- do we want to feel?
What places can we eat that are affordable, interesting, and tasty?
If I were a teenager, what would fascinate me the most about NYC?
How can we increase the depth of this experience? How can this become a meaningful trip? A life highlight?

Gritty details needed:
Addresses -- museums, oddities, eateries
Fees -- subway, museums, entrance fees
Routes -- subway, bus, walk
Hours -- historical sites, museums, daylight
Timeline -- one full day, four nights

Categories of information:
Eats--variety is the key, price is critical, atmosphere is secondary
Treats -- off the beaten path spots (carousel in Central Park that is in Catcher in the Rye)
Feats -- activities (like a show on Broadway) and tour (MoMA)
Completes -- sites and stories that fill in the holes of why NYC is incredible