I don't know why almost everyone on this forum always suggests using Clinton to get to Union Station, especially considering that most people will have luggage.
Given, it's the closest L stop. But, as noted by others, not the best suited for travelers.
The old joke in Chicago is that if you ask five people for directions, you'll get five different ways of going someplace. Everyone has a best way of their own.
That said, I'll give you the best way from O'Hare to Union Station!
Take the Blue Line. You won't stand out, as a traveler, at all. Lots of other people, including numerous flight attendants, will also be riding and hauling luggage. Yes, you might need to tuck a bag in by the seat or in the aisle next to you. Everybody does this. It is reasonably safe riding the L. Thousands of people do it every day. Travelers are no more susceptible to attack. Though, someone walking through cars with a sob story seeking money from passengers might consider one a slightly better mark. Just ignore them or politely wave them away.
Alight at Jackson. Take the elevator to the platform level. Then the elevator outside the paid area to street level. You'll be on Dearborn, outside the Federal Courthouse. Walk a half block to Adams. There you can board numerous buses, all of which will take you across the street from Union Station. (Just ask the driver to alert you to the stop, if needed.)
This is easy peasy.
Oh, and since I see that you're going TO O'Hare...
First, stop by the ticketing area for Metra. There, you can purchase a single use CTA ticket (good for the first ride and two transfers within two hours of initial use.)
Now, head up to the concourse level via the escalator by the south (even numbered) tracks. (I believe there is also an elevator near there, which may allow you to get to street level, avoiding the escalators altogether. But, I always use the escalator.) Walk ahead, staying to the right, on the mezzanine level. Then, take the escalator you'll encounter up to street level. Exit Union Station.
Some guy may offer to hold the door for you or assist with your bags. He's hustling for tips. (I'll always accept the open door, but decline any bag help. "I've got them, thanks!")
Enjoy a brief view of the Chicago River, if you like. Or, gaze up at the Willis (formerly Sears) Tower, for a moment.
Now, cross Jackson, at the light. You'll find the bus stop is right there, just steps away. Take any bus. Any bus other than the 151 will let you off at the corner of Dearborn and Jackson. Cross Dearborn and Jackson, then walk ahead (north) a half block along Dearborn to the Blue Line elevator.
Ta Da!
Now, a minor variance. If the bus you catch happens to be the 151, it will turn on Dearborn. Not to worry. It actually will drop you at the next stop right by another entrance to the Blue Line. However, there is no elevator there. So, you'll have to walk back (south) a block.
Others take note: Of course, these buses will also get you within a block of the Red Line, Brown, Pink, and Orange Lines, Purple Express, with an easy transfer to the Green Line, too. You can transfer to other north/south even east/west routes easily on Clark, Dearborn, State, and Michigan. The 151 will take you directly to the Chicago Theater, the Magnificent Mile, the Lincoln Park Zoo, and all kinds of other interesting locales on the north side. The 28 will take you to the Art Institute, Symphony Center, Grant Park, Museum Campus, University of Chicago, Museum of Science and Industry, and other points south.
