Petey's on Rt1A Rye for seafood, the best thing on the menu is the seafood chowder (not the clam).
On Rt1 in Hampton, Ron Jillians for pizza and Italian food.
Best pizza in the area is downstairs at the Portsmouth Gas Light - located downtown in Portsmouth. Flatbread is good, but it's "healthy" pizza, not your typical kid pizza.
Lago's Lone Oak Ice Cream on Rt1 for da-best damn ice cream in the area. Lago's makes their own ice cream, many of the other stands in the area sell Richardsons ice cream, not that it's bad it just isn't Lago's.
Molly Malones in Portsmouth on Pennahallow Street has a great and reasonable selection of sandwiches and burgers, and I have found it a great place to take the kids.
The Rusty Hammer on Market Street also has a good inexpensive menu.
I don't eat pizza, so I don't have any recommendations, except to say that Sal's Just Pizza is always pretty popular (I think there's one Ocean Avenue at Hampton Beach).