@yourmomma post is great advice. Follow it. I am a Chicago native and have worked in chicago offices for decades with hundreds who use public transportation. “The train” does NOT mean the " el". The only time the el is referred to as a " train" would be on a traffic report but ONLY when proceeded by the line. “Red Line Trains are experiencing delays”. But if you say " I have to catch my train" you are not talking bout the el.
@GloriaVaughn I am not sure what you are saying. You originally said named streets run north/south which isn’t correct and could really mess up a new visitor. They run all directions. And numbered streets are only South of Madison Street and not relevant to the north side.
And again, telling a new visitor that there is no “subway” will really confuse them when they have to walk down stairs under Dearborn and State Streets, especially when they can also see the elevated lines. Give the OP’s destination is the north side, they may have to transfer from the orange to the red line which is under ground at that point. It’s a long wait on the elevated tracks. 
And in all my many years, I have never heard the “L” referred to as a “train” on a regular basis. People say “we are catching the L,” or the Red line, blue line, etc. If they say train, they are usually referring to Metra. If someone approached me and said “how do I get to the train” I would have to ask several questions to figure out where they where going and what they wanted. If they said "how do I get to the “L” I can show them the closest station.
Yes, when we were tourists in Chicago, we caught the “L” and loved it. It was faster and easier than most other forms of transport, especially during rush hour. We never called it the train nor heard anyone call it that.
Depending on where you are going on the Near North Side, taking the Orange Line into the Loop from Midway and then some other train to near your destination could be really fast, really convenient, and really, really cheap (especially if you get a three-day pass and are travelling around the city enough to use it). I don’t have any idea why someone wouldn’t ride the Orange Line from Midway at night. Nothing sketchy has ever happened to me on that line. It stops at a bunch of places that register as Nowhere to me, then it’s in the South Loop, then the Loop.
If you’re familiar with the system and Chicago in general, then fine ride at night. You know what you’re in for. But for someone new to the area, with no familiarity with the “L”, riding at night, your first time, I would not recommend it. Lets face it. It’s public transportation. In a city with some problems. I know people who have had “incidents” while riding.
edit -
Google maps will give you the best routing when you arrive. I’d just take a carry on if possible, there are stairs to go up and down.
@yourmomma, thanks for your post. I am one of those who has had “incidents” - pick pocketed despite being a daily rider who was “aware” of my surroundings.
Thank you all for the information. We will take Uber—just wanted to make sure I could get it at Midway. My cousin and husband lived in the Chicago area and recently retired to Nevada. We are letting them do all the planning—we’re just along for the ride.
@Hanna "Wha – ? What neighborhood on the south or south west side doesn’t have numbered east-west streets? "
I never stated that the south or south west side do not have numbered east-west streets (see original post). The comment was in response to the comment of numbered streets are on the south side. That is true only on the far east side.
@yourmomma " And numbered streets are only South of Madison Street and not relevant to the north side."
When did Midway airport move to the north side? It is the airport that they are using which is on the south side.
@GloriaVaughn They are staying on the north side. The fact that there are numbered streets on the south side isn’t relevant for them. And the only numbered street that matters anyway is 35th street.