Brookfield Zoo, reached on the BNSF line out of Union Station.
http://www.brookfieldzoo.org/czs/flash/landing/index.html
Chicago Botanic Gardens, reached on the UP-North line (Braeside station) out of Ogilvie Station.
http://www.chicagobotanic.org/
Arlington Park horse-racing track, reached on the UP-Northwest line out of Ogilvie.
http://www.arlingtonpark.com/
Ravinia outdoor concert venue, reached on the UP-North.
http://www.ravinia.org/
Minor-league baseball in:
*Joliet, at the end of the Rock Island line out of LaSalle St. Station.
http://jolietslammers.com/
*Schaumburg, reached on the Milwaukee-West line out of Union Station.
http://www.boomersbaseball.com/
All of these are right across from the station, except for the Zoo which is an easy walk away and the Botanic Gardens which are walkable but not as close as the Zoo is to its station. You'd be doing a lot of walking at the zoo and the gardens anyway. :^) For something a bit more ambitious, there's:
Indiana Dunes state & national parks, reached on the NICTD (formerly South Shore) out of Millennium or VanBuren St. stations
http://www.indianadunes.com/
If your family is into bicycling, there are plenty of suburban bike trails reachable by Metra. (Of course, in the city there's the spectacular Lakefront Trail
http://www.choosechicago.com/articles/view/The-Lakefront-Trail/454/ and some others.) The (literally!) original rail-to-trail is the Illinois Prairie Path, which can be reached from several stations of the UP-West line out of Ogilvie.
http://www.ipp.org/
Of course, there are some nice suburban downtowns near Metra stations. Lots of interesting ones on the North Shore, reached on the UP-North line.
If you don't mind that only a VERY limited number of trains go north of Waukegan on the Union Pacific North Metra line(even on weekends too, sadly), Kenosha is a very fun place to travel to! I'd recommend going on a Saturday when you can get a weekend pass, explore one or more(your call if you only do 1, or more) of the 'north shore' downtowns south of there, then head north to Kenosha. There are a lot of good restaurants and bars up there, it has the free Kenosha Public Musuem(basically their local natural history museum a la The Field Museum in Chicago, plus also a civil war museum and a kids museum), you can ride on a restored streetcar loop through the downtown(which runs till something like 6:15-6:30pm), it has a farmer's market if you go early up there(HarborMarket), and of course the lakefront is really nice to see(i.e. Simmons Island beach, the lakefront park on both sides of the harbor). Don't forget that the downtown branch of Kenosha's library was designed by Daniel Burnham(of Rookery Building, Chicago Union Station, and Flatiron Building in NYC fame. and dozens more buildings at
https://en.wikipedia.org/wiki/Daniel_Burnham). Across the street too(and in nearby blocks) is a nice historic neighborhood of single family homes, which even includes the home Orson Welles(yep, that one of Citizen Kane movie fame) was born in. This blue, white, and red brick home is it, on 7th Avenue just south of the block the library and park are on: https://goo.gl/maps/xA7uPBtBNmx
Anyway, the Metra Union Pacific North schedule. Where you can how MUCH fewer trains run to Kenosha vs. those that begin/terminate in Waukegan, but it's still very fun to visit on an $8 weekend pass:
https://metrarail.com/sites/default/files/metra_50933_fm10_tt_proof_1.pdf
I also would've thrown downtown Geneva in(Union Pacific West line), as an interesting place to visit. You can railfan in nearby West Chicago(also having a UP-West stop, and remember all you need is an $8 weekend pass), at this railroad junction you can walk to from the West Chicago Metra station. Here's a google street view preview of what I mean:
https://goo.gl/maps/xQzABvyoXPB2
Also a shame Blue Island, IL wasn't mentioned. It has a pretty historic downtown, and is just south of Chicago(and the Beverly neighborhood) on the Rock Island Metra line. Has a lot of interesting restaurants and bars on both sides of the Cal Sag Channel. More just to the south of that bridge, but there are a few to the north(the side of the Metra station) as well. Also has a few antiques stores and vintage shops, too. If you walk over south of the Cal Sag Channel and a few blocks west on Broadway Street, you'll run into an AMAZING place for railfanning(another google street view preview):
https://goo.gl/maps/G9twmyZafeT2
Speaking of places south of Chicago, Flossmoor, IL has a nice downtown, complete with also an interesting restaurant and beer brewery right next to that station(Flossmoor Station). Ditto with Homewood, on that line. Also, it should be mentioned for the South Shore Line, that Beverly Shores and downtown Michigan City(11th Street, it's the first Michigan City stop of the 2 when you go east from Chicago) are nice day trips to do. Be cautioned that Beverly Shores is a flag stop, and you just press the signal button on the east end of the platform 5-10 minutes before the westbound train time for the arriving train to stop. And of course when you're heading east towards that station, ask the train conductor/employee taking your ticket which cars of the train will open at Beverly Shores, since not all train doors open. Michigan City has a nice little downtown, bunches of restaurants/bars you can eat or drink at, Washington Park and Beach, and even an outlet mall. There's even a casino(Blue Chip) that has a shuttle from the MC 11th Street station, as well. And don't forget(for railfans), the VERY last section(since East Chicago and South Bend got rid of their street running sections decades ago) of the South Shore Line that has street running! Amtrak runs through the downtown as well(most blow right through MC, but a handful of the Michigan trains stop here), so you might catch a train or 2 going through Mich City.
If you have the money and desire, I hear Rochelle, IL Railroad Park is a action packed place.
I've been there once, and liked it. Definitely do need a car to get out there though, or be up for doing a bike reservation on Amtrak to Mendota, IL, then biking north and back(2.5 hours each way) to Rochelle.
I want to thank everyone for the advice on the Illinois Railway Museum. I did rent a car from Hertz which I was very disapointed with. However, I did enjoy my time at the museum. They had one trolley and a main line train running. I just did not have enough time in the afternoon to ride the train. They are working on getting steam running in the near future. There is alot to see there. It was not as "polished" as some other museums I have been too. The reason I say that is the equipment is just stored in pole barns, with tight aisles, and dimmly lit. However, they are a very active museum and it looks like they are doing as much as they can to keep stuff inside. I was dispointed that the trolley buses were not operating.
I need to revisit IRM, someday. Too bad they don't have a complimentary shuttle service from the Crystal Lake Metra station, since that'd for sure probably boost their attendance(at least from those who don't own cars) a little bit.
