I would stay in the North Michigan Ave area. I've stayed at the Wyndham, the Millennium, the Monaco and the Courtyard. You may want to try Priceline, as Chicago hotels can be pretty pricey. Although, doing research for a trip that I took about a year and a half ago, I found the Sheraton Four Points to be reasonably priced and had great reviews on Trip Advisor. We ended up with an employee discount at another hotel so did not stay at the four points. But it did look really nice, for a good price.
You definitely will want to hit the Science and Industry Museum, but it's a trek from downtown. You'll either need to take a cab or a bus, so it may not be worth it to you. The Aquarium is also fantastic. The Field Museum is supposed to be great too, if you like that sort of thing. I haven't been since I was a kid, but I remember hating it.
There's also Navy Pier which has a lot of cool things. If it's not too cold and if it's even running, check out the Sea Dog speedboat. There's a children's museum there too, but I haven't been to it. And there's also a cool maze house.
One of my favorite things is the Observation Deck of the Hancock building. Or the Sears Tower would be cool, I'm sure too. We always stay near the Hancock bldg so that's why we go there. But next time we'll likely try the Sears Tower because my son thinks that's cool.
Oh man, now you're giving me the itch to go back to Chicago.
As for the train, I would recommend you get on the amtrak site and do a "status check" on your train, for the last few days. A girlfriend and I were going to take the train from STL, but found that it was consistently late nearly everyday. Not just an hour late, but sometimes 6 hours late. So we decided against it. We found the same problem about 8 years ago too. I know the train sounds like a fun easy idea, but if you're going to get a late start to a short trip, that could really put a damper on things. Hopefully, your route isn't so bad.
ETA: Try Uno's for pizza (several locations) and Ed Debevics for burgers. It's a fun 50s diner. I wanted to go there last time we took the kids and we didn't make it, to either. Man, I'm really feeeling the need to get back to Chicago now.