You'll be going to L stations both ways, which do not take cash. If you have a Ventra card from previous travels, you can reload it. If not, best bet is to use a contactless card or Apple/Google/Samsung Pay. $2.25 at Clinton, $5 at ORD.
Of course you can pay with cash at L stations (and on buses, too). A Ventra Card is not required.
When you get to the L station, there will be a large silver vending machine called a "Ventra Vending machine." Just buy a single-ride ticket from the machine (cash or credit) as demonstrated in this video:
You'll note that in the video, the actor pays in cash.
Then take the ticket and go to the turnstile and pay like this:
A single ride ticket costs $3 except at the O'Hare station where it costs $5.
The closest stop to Union Station is the Clinton stop.
Clinton is a one-way street that runs along the west side of Union Station. If you exit Union Station through the Great Hall, you'll be right on Clinton, but you can use any exit, you may just have to walk around the building.
Walk south on Clinton. Clinton is a one-way street, south is the direction the cars are going. In about 2 blocks you will get to a highway bridge going over Clinton. The entrance to the station is under the bridge. Note that there is no elevator or escalator for the first set of stairs. After you get down to the payment level there is an escalator that works most of the time.