First, welcome to CT Cougfan. I hope you have a safe trip on your way here! If you're making a weekend of it and have some time to do a little driving, then I would recommend going to either New Haven (down by the shoreline) or Middletown (about 20 minutes south of Hartford). Both towns have a very wide array of all sorts of cuisine that could appeal to your whole family.
New Haven - Pepe's Pizza is probably the best in the state. The lines get pretty long, especially on the weekend,so I'd recommend going sometime in the mid-afternoon early evening. Sally's is Pepe's archival (the Geno's vs Pat's Philly Cheesesteak battle of New Haven pizzas).
Middletown - Taino Smokehouse is the best BBQ in the state. You will most likely be unimpressed with the location and curb appeal (it's in a strip mall next to an Ace hardware store), but the meats are smoked to perfection. The sides are delicious. Do yourself a favor and order a pulled pork sandwich. Other Middletown treasures: Eli Cannon's (great selection of draft beers and American cuisine), First & Last (pizza, italian, family friendly), and O'Rourkes (small diner, GREAT breakfasts).
It's too bad your family isn't in to seafood. If you get adventurous and adopt a "when in Rome/CT" attitude, take a 30-40 minute drive down to Westbrook, CT and go to Bill's Seafood along Route 1. Best seafood in the state and your kids can feed the ducks and seagulls from your outdoor seating table. There's even an ice cream stand right outside if you guys can still fit it in after your meal! The CT shoreline is right there so you can grab some cones and then head down to the water and overlook Long Island Sound.
On game day, be sure to walk through the blue and red lots. Some chatter you hear out there won't be kid friendly (Friday night games tend to make the brews flow much easier at our stadium), but our fanbase is, for the most part, very friendly and hospitable. If you can't find a single hospitable fan, go to the first blue lot on your left when you enter the Silver Lane entrance (the "Pratt & Whitney entrance"). About 10-15 cars down on the right along the woods you will find me. We are serving pastrami burgers, which I'm told by
@huskymedic and others is a Utah staple. I'm bringing enough to feed 8 or so and I think our group is going to be 4-5. I'll be in a blonde mullet decked out in my Husky blue.
Enjoy your trip out east and the game!