Not a native Chicagoan, but we've stayed at the Holiday Inn in Evanston (right by the purple line) several times while visiting Chicago. It's south of NW's campus (about a 10 minute walk IIRC). Usually pretty good rates there or at least they were in the past. IDK about the trains (Landlord or suh_fan can maybe weigh in) but you'd probably want to Uber to Ryan at the very least.
Yeah HuskerN you could just look for a place in Evanston if you can like Enhance did since your going to catch the Northwestern game there anyway. You can take the Purple line straight south into the city from Evanston also. On the flip side if you want to stay in the city and then head north to the game try to find an airbnb in a neighborhood along the lake (Lakeview, Lincoln Park just to name a few). Lakeview has a ton going on in terms of bars and restaurants and Lincoln Park the same but both offer train access north up to Evanston. I hope that helps a little and hope you have a great time!
Lakeview, Lincoln Park, Wrigleyville and Andersonville are all lively neighborhoods off of the red line if you want to stay in them or go out when you're not at the game. Wrigleyville will probably be the most packed with Nebraska fans the Friday beforehand.
Another option would be to get a bus ticket from Kirkwood Bar. I think $20 gets you one or two free drinks and free shuttle bus service to and from the game.
Can I take a train from the 'burbs to places like the Shedd Aquairium & Adler Planetarium? What about the Field Museum? Or do I need to stay closer to downtown?
You can, it just takes a while. As far as CTA trains (what everyone calls the 'L'), the red line runs way out of the city both north and south and is a straight shot through the heart of the city (probably a 15 minute walk to the Field/Shedd museums from the Roosevelt stop), but most all of the lines run out into suburbs somewhere and all still head down to the loop downtown.
There's also METRA trains that run all over the northern part of the state, but I've never actually used one of those.