If you're not raising your children in the Buffalo Niagara region, then you're a bad parent. Take that, all you parents from Kansas City!
Jen and Alan provide details on the great things there are to do with kids in this area. I have some favorites, myself.
In terms of restaurants, Buffalo has a ton of family-friendly eating establishments. Favorites of ours include Pizza Plant in Williamsville (where, in addition to crayons, the kid can get a wad of pizza dough to play with while waiting for food) and Taffy's in our own hometown of Orchard Park. Taffy's is just your basic burgers-and-ice cream joint, but it's situated on one of the Southtowns' busier intersections, so it makes for good car-watching, and its seating consists of a whole bunch of those covered picnic benches that "rock" back and forth on two tracks. Fun place.
In terms of Buffalo shopping malls, the Galleria is the big one, and it's got plenty of stores of interest to kids. The Boulevard Mall in Amherst is a bit better for small kids, since it has a carousel and an enclosed play area that has a bunch of play-items made out of this shiny-rubber stuff. This area in the food court is always packed with kids. (Of course, none of Buffalo's malls approaches Syracuse's Carousel Center, which is still the best mall I've ever been in short of the Mall of America itself.)
The region's best-kept secret, as far as free places to take the kids is concerned, has to be the Charles Burchfield Nature and Art Center on Union Road in West Seneca. This is a wonderful little park, partly dedicated to the art of Charles Burchfield (a prominent water-colorist who lived in West Seneca for a time), a beautiful set of park slides, and a set of nature trails that wind back into some surprisingly dense woods and down to the side of Cayuga Creek, whose water is generally shallow enough for wading (in late summer, the water gets so low you can wade all the way across and not get wet past your knees) and swift enough to be fun to throw or skip rocks. This is one of our absolute favorite spots in the area -- and it's completely free to go there. Give it a shot, folks. On a warm day, bring a picnic lunch. You'll be glad you did.