Fairway is fantastic - an legendary grocery store, you'll be able to get loads of stuff there. Citarella, next to it, is a bit higher end but also has good prepared food.
A few suggestions and I'll try to come back with more after I've had a think/have more time
The 1,2,3 subway line runs up and down the west side and is very convenient - below Central Park, it's pretty easy to walk across town from most stops. Manhattan gets narrower as you go downtown, so the walks get shorter, although the cross town blocks are longer than the ones that go up and down. It's basically a big grid system, so very easy to navigate.
For old fashioned southern food at a reasonable price, Miss Mamie's Spoonbread Too is fun - It's nice to go up and have a wander around Morningside Heights, where Columbia University is, and have a meal there. Also worth a visit - The Cathedral of St John the Divine and The Hungarian Pastry Shop - a true old school cafe hangout for Columbia students and faculty. The Met Cloisters, a bit farther up, is a lovely museum.
I love Daily Provisions (a small chain) at Amsterdam & 78th for breakfast sandwiches.
Even if you don't want pizza every day, make an excursion down to Johns of Bleeker Street for one. I'm also a huge fan of the square pepperoni slice at Prince Street Pizza (no seating, so it's lunch on the go).
Zabars is worth a walk-in. Their bagels and bagel sandwiches are good, although the bagels at Absolute Bagels at 108th are better.
For Chinese food, I haven't been to this place in years, and I think it was closed but recently reopened - for delicious and very cheap food, we used to swear by Great NY Noodletown at the corner of Bayard & Bowery. It used to be kind of an institution - total no-frills but a real hangout for some of the most famous chefs in NY. Their shrimp wonton soup was amazing! And around the corner, Chinatown Ice Cream factory for fun, unusual flavours.
Pancakes - truthfully the best restaurant ones are at Chez Ma Tante in Brooklyn. The neighbourhood (Greenpoint) is a cool one to walk around if you're wandering over to the really trendy Williamsburg, but probably not worth going just for pancakes! Closer to where you're staying, Good Enough to Eat at Columbus & 85th does reasonably good ones.
If you have hot dog loving kids, try Gray's Papaya at Broadway & 72nd. Another NY institution, it doesn't get cheaper than that (although full confession, I haven't personally had one since I was about 18).