The Museum of Natural History is excellent. The Metropolitan Museum of Art is also fantastic.
If you want the best fucking Italian food in that city, go to Da Tommaso on 8th Avenue between 53rd and 54th. Jerry Orbach from Law and Order used to go there all the time. I've been going there for years, its far and away my favorite restaurant in the city.
Another good place, if you're looking for simple Irish food or a good Irish pub, is The Playwright. There's two of them, on is on 49th I think around Times Square, the other is on West 35th. They're both great places, good food, not too expensive, good Irish atmosphere.
Another cool place is Fraunces Tavern down in the bowery on Pearl Street. If you like history, it's where Washington gave his farewell address to his officers, and it's a restaurant downstairs and has a cool museum upstairs.
Any of the bars in midtown will give you solid Irish fair - Pig and Whistle, Scruffy Duffys (which may be closed), etc. There's a place Red Sky on east 29th that's a cool bar also. My buddy's cousins work there and I got very drunk there some years ago on St. Patrick's Day. It was a great time.
There's a French restaurant in midtown also called Pigalle on 8th avenue. They're a little expensive as I recollect but their desserts were fucking fantastic, it's worth the trip and it's real close to Da Tomasso also.
As far as sightseeing and shit, there's no end. You can all wander around Times Square and take pics so the people like me fuckin curse at you because you're not paying attention, you can go to the massive Toys R Us that's pretty awesome, the Hershey store, and there's a lot of fancy ass stores like Coutore and all that around there that you'll get dragged into if you've got you woman with you.
Rock Center is up from midtown, it's not spectacular if it's not Christmas, but if you go in the winter it's absolutely worth it for the atmosphere alone.
Many of the small places you'll amble into are pretty good, from the corner pizza places to the burger joints...another good place is called "Big Nick's" and it's on Broadway and 77th. Great burgers there, cool little place with all pictures on the walls and shit, real old time place.
Another cool place was John's Pizzeria on West 44th. It's an old church that was converted into a pizza place, makes for an interesting look. The pizza was pretty good I think but it's been a long ass time since I been there.
For the dude that said he's going in July, have fun, it's hot as fuck here.
St. Pat's Cathedral is on 5th Avenue and 51st. It's really one of the most gorgeous churches you'll ever see. I always stand across from it when I go to the St. Patrick's Day parade cause the sun is coming down on the other side in the morning.
My long rambling post is done for now until I think of some other worthwhile places in the city, of which there are many.