Visit NYC with a couple of 7 year Olds
Since you're in Brooklyn Heights, I would recommend Franny's in Prospect Heights and Lucali's in Carroll Gardens. Franny's will be easier to get into, partly because it's a bigger space, but you can always do take out from Lucali (and Franny's). In Manhattan, Lombardi's is good and I know people who attest to Artichoke (which now has a few locations), but I prefer Lombardi. If you're feeling adventurous, check out Roberta's in Williamsburg.
32 Spring St, New York, NY 10012
328 E 14th St, New York, NY 10003
Another suggestion is dim sum in Chinatown. If you prefer to avoid the crowds in Chinatown, another good option is Chinatown Brasserie in the East Village. There's also Red Farm by the same dim sum chef from Chinatown Brasserie - guess he's doing double duty - which opened not too long ago. Haven't been yet, but I'm guessing it's good.
380 Lafayette St, New York, NY 10012
529 Hudson St, New York, NY 10014
Do you want Brooklyn recommendations or Manhattan ones? For Brooklyn, you need to post on the Outer Boroughs board.
Regarding "best pizza in NYC" -- how far are you willing to travel & how long are your 7 year olds willing to wait?
Some of the best pizza in the city will be far away from Brooklyn Heights and you're likely to face a wait once you get there.
Really, it depends what you're looking for: gas oven style, coal oven style, Naples style? A slice? A pie? Takeout? Sit down? Or is standing up OK with you?
There are a few distinct styles of round pizza found in NYC: New York gas-oven style, Neopolitan style, and a hybrid style of the two that is also unique to New York (usually coal oven). Then to throw another wrench into things, some places are known more for square pies (like Artichoke).
Note that lot of famous places like John's of Bleecker, Grimaldi's, and Lombardi's are pies only. Hard to make a recommendation without knowing what you're looking for: whole pies or slices, gas oven vs. coal oven.
If you are limiting yourself to only Manhattan, my favorites, agnostic of oven type:
John's of Bleecker, if you ask for it well done. Get it plain or with one topping, max. I'm partial to their green peppers. This is classic coal-fired NY-Neopolitan hybrid style pie. They do sometimes undercook/under char it, though. Whole pies only.
Motorino for Naples style. Delicious but not really historically "New York" style. Crimini and sausage, spicy soppressata, or whatever their special pie is. Wonderful crust, quality toppings. Whole pies only.
For both you may have to wait in line. I have also enjoyed Patsy's in East Harlem (coal oven) in the past but it is a bit far uptown dependent upon where you are starting from, and I've not been very recently. Whole pies OR slices if you want.
South Brooklyn Pizza or Joe's for a slice (gas oven). Joe's is a bit less crisp/more chewy and on the more cheesy sied and has a more uniform appearance. South Brooklyn is more crispy and has an interesting cheese blend (mozzarella, grana padano, and fontina) with fresh basil, and the cheese and sauce are more scattered, which you usually don't see at slice joints.
If you are willing to go outside of Manhattan, Roberta's is good but I'd go for lunch, not dinner, as waits can be long for dinner. I love Franny's too and they are open all day on weekends, so go for lunch or very early dinner (like 5pm) to avoid waiting. The waitstaff are super sweet to kids, too. I'd avoid Lucali unless you have VERY patient 7 year olds.
Another one that kind of flies under the radar is Best Pizza in Williamsburg. It's wood fired but comes out tasting more like NY gas oven style somehow.
My real OB favorites are Di Fara and Totonno's Coney Island but they will be a trek from Brooklyn Heights and at Di Fara you'll definitely have a long wait and may need to fight to get a table.