From O'hare Airport, jump on the 'L' train towards Downtown Loop. Transfer in the loop from Blue to Red towards Howard and get off at the Chicago stop.
Walk a few blocks east and one block south to Superior to get to your hotel, MileNorth, an LGBT-friendly hotel in a prime location.
After dropping your bags off, take a cab to nearby Navy Pier, a tourist destination with tons of food, shopping and the famous Ferris Wheel that never stops moving. The Children's Museum is also nearby.
If you walk to the bus terminal on the north side of the Navy Pier entrance, you can jump on a bus (either the 44 or 54) to get back into the city area. Ask the bus driver to let you know when to get off for a Blue Line connection, and then transfer to an "O'hare bound" L train to arrive at Logan Square.
Lunch at Lula Café, across the street from the train station, is next on your family's list. This LGBT-friendly, farm-to-table restaurant has a variety of local and sustainable offerings to choose from, including incredibly tasty cocktails and artisanal coffees.
A 15-20 minute cab ride can get you to Lakeview, aka Boystown, where you'll find the Center on Halsted, an LGBT community center with a massive calendar of events for a wide range of needs and wants. Check the calendar of events for family-friendly movie screenings or special events!