The Notebook at Chicago Shakespeare is pretty swell and Navy Pier is nice if your visit is short.
The show currently on Second City's Mainstage "Do the Right Thing, No Worries If Not" Is one of the better ones in recent memory. If sketch comedy isn't your thing, obviously avoid.
Clyde's is the same production (including some of the actors) that appeared on Broadway. I think it plays a bit better now but if you've already seen the production you don't need to see it again. Unless you want to I guess.
Souther Gothic is sort of an alright play but the production is immersive and fun. You wander around a house with the characters at a dinner party and follow the stories you want to. It's right in the loop (adjacent to The Goodman) so it's very accessible if you're staying in the area.
The tiny production of Sweeney Todd in the basement of the Chopin is a delight. If I hadn't seen the pie shop production a few years ago I'd say it was a real revelation. Even if you did see that one... it's worthwhile.
There's two world premiere's that show real promise. Swing State at the Goodman. Rebecca Gillman in a political mode with two of our best actresses, Mary Beth Fisher and Kirsten Fitzgerald and former Broadway Scorpious, Bubba Weiler. At A Red Orchid, Brett Neveu premiere' a "sequel" to the Magnificent Ambersons, The Malignant Ampersands. Since it hasn't played a performance yet, I can't speak to the quality of the script but A Red Orchid's shows are usually as intimate, intense, and Chicago as things get.
Unfortunately, Fiddler will be closed before you arrive.
Jesus saves. I spend.