I like what daveklepper says. Same idea as mine but a much better location. Plus you get to ride the South Shore.
I took the Empire Builder from Milwaukee to Portland last December. I took the Texas Eagle from Dallas to Chicago and a Milwaukee Service train from Chicago to Milwaukee. I stayed over night in Milwaukee, as opposed to Chicago, because the cost of a hotel room in downtown Milwaukee was much cheaper than a similar hotel room in downtown Chicago.
You might check out parking at the Milwaukee Airport. If it is anything like most airports, the long term parking would probably cost $6 - $8 a day. Amtrak has a station that serves the Milwaukee airport. You could drive to the airport, park the family buggy in long term parking, and take a Milwaukee Service train into Milwaukee, where you could catch the Empire Builder.
I live in Georgetown, Texas. When I want to take Amtrak from Austin, I drive to the Austin Airport, where I park for $6 a day. Parking at the Austin Amtrak Station is a bad idea, unless you have an old car that you want the thieves to take off your hands. Then I take a city bus to the Amtrak Station, where I catch the Texas Eagle. If the train is late returning to Austin from the north, I catch a cab to the airport. I don't like being downtown in any Texas City late at night.
I don't think you are going to find anything cheap in Chicago near Union Station. There might be parking at or near a Metra station. Most Metra trains arrive at Union Station. Guess your other option is a bus. What about a train from the Detroit area? Best of luck.
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