Just to clarify. We could be talking about 3 different kinds of trains.
1. Amtrak. You can take from Lincoln/Omaha directly into downtown Chicago, Union Station.
http://tickets.amtrak.com/itd/amtrak
I've taken Amtrak from Lincoln to Chicago. Not a bad trip, just a little long. Seats are comfortable enough to get some sleep.
It looks like there is one train that leaves on Saturday from Omaha and Lincoln and gets in at 2:50 pm. If it is a night game you can get off at Union Station walk a few blocks east and take the CTA purple line up to Evanston. If it's a noon game it might be a little awkward. You'd have to stay overnight on Friday and take Friday off from work.
2. Metra. Trains that run from Suburbs into Downtown Chicago. I believe most routes end at Oglivie Station.
http://metrarail.com/metra/en/home/maps_schedules/metra_system_map.html
If you have friends that live in the suburbs you can take the Amtrak, walk across the street from Union Station to Oglivie Station and visit them. When coming into Oglivie it's useful to know which street you want to head toward since the exits are marked by which street to exit on. The nightly routes are spotty so if you want to party until 1:00 but need to get back to the burbs, you might have to wait until 5:00 or 6:00 before the next train.
3. CTA. The bus service also goes by this name. Runs local routes throughout Chicago. Been a while since I took this but I believe you need to purchase a card to get on and off. There are vending machines at the station so keep some cash handy.
http://www.transitchicago.com/riding_cta/systemguide/default.aspx
You can get from Midway or Ohare airports on it.
http://www.transitchicago.com/riding_cta/airports.aspx
If you scroll down to the downtown map here and zoom in you can see where the purple line is relative to the Oglivie and the Union Stations. Don't doze off if you need to take the Red line since you could end up in South Chicago.
http://www.transitchicago.com/assets/1/brochures/ctaSystemMapBrochure.pdf
If you get drunk and disoriented and can't find the CTA line, running short of time or whatever, you can always hail a cab and tell them to take you to Oglivie or Union Station. Most likely their english won't be very good but they'll get you to those destinations.
Omaha to Evanston looks like 7 1/2 or 8 hour drive so theoretically you could get up early early in the morning and make it to a noon game. If you can do it, the better option would be take off work early on Friday drive and stay overnight. That's what I usually doing going the other direction. Sleep in a little on Saturday and be ready for the noon game. It's a drive I've made too many times. The windfarms in western Iowa are fascinating, otherwise get used to looking at corn and soybean fields (and it's Iowa). There is the Iowa 80 truck stop, apparently the worlds largest if you believe their sign. Food's not bad there. Gas in Chicago is more expensive so fill up before you get close.
Here's information on parking and transportaton to the campus
http://www.nusports.com/tickets/gameday-info.html
People in Chicago do have a thick accent. Try not to laugh. Also, they can be abrasive and rude so don't be shocked, it's normal. In case you haven't figured it out yet, Midwesterners aren't quite the same as people in the Great Plains states.
Now I'm getting excited for this game. I need to get a ticket.