Classic NY Pizza, Bagle, Deli
Each of these areas are threads onto themselves and if you do a search you will find these three topics are about as divided as most in NY.
The jfood picks:
Bagels - I like Ess-a-bagel on 3rd and 51st. It's great and close to my NY office, but cannot compare to the others that will be mentioned.
Deli - OK one word - Katz. If you do not want to go to LES and that would be a mistake because of all the other great food and heritage in the 'hood, my other uptownplace is Carnegie. Others may disagree but jfood thinks the food is great and jfood loves hearing the "OMG do you believe how big this sandwich is". Especially like the look on the face of the big guys who order a double and then just stare. Jfood only negative is after 20+ years going to Carnegie, ordering my pastramion rye from an Asian waiter is not the same as the Borsht Belt Imitator you grow used to.
Pizza - Jfood like John's on Bleeker. It's a comfort zone thing like Sallys in NH versus Pepes. Maybe its and old dog new trick analysis (see avatar).
I second (or third or fourth, whatever it's up to) Grimaldi's as the reigning champion of old-NY style pizza. Well worth the trip. Only the 2nd or 3rd best pizza in the city, but the absolute best of the classic NY style pizzas.
Another great option that keeps you in Manhattan is Patsy's in Spanish Harlem, but Grimaldi's is probably easier to get to even though it is in Brooklyn. Both places have a great coal oven char, a crust that is crisp in places but soft in other places (foldable without cracking), excellent ingredients, outstanding atmosphere... I could go on at some length about these places.
Arturo's is a cool place, and their pizza is pretty good (it does have that coal-oven char), but in my opinion it's not in the same league, although it may have them both beat for atmosphere!
Grimaldi's is especially good for kids, because of the pier with an ice cream parlor and beautiful photo opportunities about a block down the road. For the adults, though, I'd skip the ice cream and get a cannoli - top notch!
Nick's is very good, but I'll probably never be able to eat pizza there again - the calzones are too good! They have ASTOUNDING cannoli, too. But the pizza, while very good, lacks the coal flavor and crunchy chew that I think of as essential to "NY style" pizza.
Katz's is an automatic on the deli front - with 2nd ave gone, it reigns virtually unopposed, and will not disappoint.
I'll defer to the opinions of others regarding bagels, but for me they don't get any better than fresh, plain bagels from H&H. No butter, no cream cheese - just a warm, crunchy, chewy, tasty bagel. I eat them right outside the store, on the benches in the median of Broadway.
Bagels - Ess-a-Bagel is my favorite. They have a very high turnover, so there's bound to be a hot flavor or two whenever you arrive. The sesame, poppy and everything flavors are great. Murray's Bagels gets good write-ups on this site, and they're pretty good but sometimes inconsistent. For pizza, if you are by Columbia University, try Koronet. For deli, Katz's by default.
Grimaldi's in Brooklyn is my favorite...plus, there's great views of Manhattan! SUPER easy to get to from Manhattan: http://www.grimaldis.com/
My favorite "deli" is Tisserie, located in Union Square. It's not traditional, but they make an amazing turkey sandwich on their homemade ficelle bread! www.tisserie.com
Agree, LFeinberg - Grimaldi's is great. There will be a line most likely but it generally moves at a decent pace. Also, after pizza there is the Brooklyn Ice Cream Factory just down the block from Grimaldi's.
Alternatively UGA Buck you can go further into Brooklyn to Totonno's which I have to say is really quite good - and sometimes I think it's better than Grimaldi's. It's near Coney Island which has had it's better days but still is worth seeing once.
And I wouldn't do my own 'hood right if I didn't mention Nick's Pizza in Forest Hills. Where so many other places do coal ovens for pizza, they have a gas oven but it's fantastic. From Penn Station, it's 15 minutes on the Long Island Rail Road.