There are two major variations of our delicious Chicago-style deep-dish pizza: the single-crust "pizza in the pan" introduced by Uno (in 1943) and Due (in 1955), and the double-crust "stuffed pizza" introduced by Nancy's (in 1971) and popularized by Giordano's (since 1972).
If you've never had either type here in Chicago, I recommend starting with the single-crust pizza in the pan. The original Uno and Due, at Ohio and Wabash, are the closest to the Hyatt for getting great pizza of this type; they're about five blocks northwest of the hotel.
www.unos.comIf you'd like to try double-crust stuffed pizza. Bacino's is at 75 East Wacker, a block west of the Hyatt.
www.bacinos.com Giordano's is in the Prudential building, two blocks south of the hotel.
www.giordanos.comWherever you go for Chicago-style deep-dish pizza, phone ahead with your pizza order to avoid waiting 30-45 minutes while seated for your pizza to bake.
For a nice restaurant that is not a chain, there are many possibilities. Assuming your children are well-behaved, you should be able to take them anywhere without any worries. Here are some recommendations in the four categories you have mentioned.
For American, we have many excellent contemporary American restaurants. Atwood Cafe, in the Hotel Burnham about five blocks southwest of the Hyatt, is one.
www.atwoodcafe.com My favorite is Cafe des Architectes, but it's a bit further, about nine blocks north of the Hyatt.
www.cafedesarchitectes.comFor seafood, go to Shaw's Crab House, about three blocks northwest of the Hyatt.
www.shawscrabhouse.comFor Italian, Coco Pazzo Cafe is about four blocks north of the Hyatt.
www.cocopazzocafe.com Coco Pazzo Cafe is very good for what it is - an Italian restaurant that is slightly less expensive than many other mid-priced Italian restaurant. Its sister restaurant, Coco Pazzo, is even better, but it's further, about ten blocks west of the Hyatt.
www.cocopazzochicago.com Also excellent is Cafe Spiaggia, about a mile north of the Hyatt.
www.cafespiaggia.comFor the best steaks in the city, go to David Burke's Primehouse in the James Hotel, four blocks north of the Hyatt.
www.jameshotels.com/Chicago-Hotel.aspx? ... PrimehouseHere's one more possibility for your nice dinner, one that is closer to the Hyatt than the places already mentioned. Aria is located in the Fairmont Hotel, across the street from the Hyatt. They specialize in pan-Asian cuisine, but they also have steaks and other American fare on the menu. Last time I ate there I had an excellent smoked prime rib. The food is excellent and so is the atmosphere, which is very classy (as you would expect from its location in a luxury hotel).
www.ariachicago.comHere are two more recommendations, for things you didn't mention but are worthwhile checking out and are not far from the Hyatt.
Fox & Obel, our premier gourmet grocery store, is very close to the Hyatt, just across the river and a few blocks east. In addition to things like the finest meats, fish, cheeses, etc that may not be of interest to a visitor from out of town, they have wonderful baked goods (don't miss their cinnamon swirl rolls!) as well as sandwiches, salads, and other prepared foods. They also have a cafe (nothing fancy, just your basic coffeehouse atmosphere) in the rear of the store where you can get anything from a cup of coffee to a complete meal prepared to order.
www.fox-obel.comGarrett Popcorn is a local specialty. They have caramel popcorn (with or without pecans or cashews), cheese popcorn, and a mix of the two. It's a great snack for when you're walking around. They have stores in a lot of locations downtown, including Michigan and Ontario (four blocks north of the Hyatt) and State and Randolph (four blocks southwest of the Hyatt).
www.garrettpopcorn.comEnjoy your visit!