hi. born and raised in buffalo! there are tons of great places to go with kids.
for a classic buffalo pasta experience, chef's is great. it's on seneca street, downtown. no frills here, but the spaghetti parm is excellent. personally, im partial to ditondo's tavern, which is just down the street, but theyre only open m-f for lunch and also dinner on fridays. mama ditondo still makes all the homemade pies and lives upstairs. delicious. i grew up in one of the buffalo suburbs and our favorite local restaurant growing up was lebro's in getzville, 330 campbell blvd. mostly known for their pastas, but they have some fantastic sandwiches and such. definitely good for kids. lots of options.
there arent a ton of fish and chip places (i cant think of any, honestly), but if youre looking for something simple, ted's hot dogs is the place to go. there are multiple locations (sheridan drive, niagara falls blvd) and while it is definitely nothing fancy, its quintessentially buffalo. john & mary's subs are also a buffalo staple with multiple locations.
of course wings and beef on weck, which have been covered extensively here. i favor schwabl's for BOW and duff's for wings. both are fine places for kids, but keep in mind that the duff's wings are hotter than most places. definitely go for the mild.
other great spots: lake effect diner (quality diner food) and romeo & juliets (pastas and delicious paninis) in north buffalo, ambrosia (greek diner) in the elmwood village. hertel ave and elmwood are worth checking out. if youre going early enough, most places are fine with kids.