Formerly known as Shakespeare Repertory, Chicago Shakespeare Theater has a beautiful space at Navy Pier. Located near the eastern end of the Navy Pier complex, the space is visible all the way from Lake Shore Drive, with a vertical sign proudly bearing the name "Shakespeare."
The theater itself is impressive from top to bottom; the lobby is an enormous construction of glass and steel, and affords a great view both of the lake and of the city. There is a sizable main stage space that is based on theater designs from Shakespeare's day; there's also a smaller upstairs studio theater. Feel free to loiter in the lobby, which contains a gift shop, bar, concessions and a complementary coat check...the theater takes great pains to ensure that patrons have a classy experience from the moment they set foot in the building.
Due to its proximity both to the Magnificent Mile and the host of on-the-pier businesses, finding dinner while seeing a show is no-brainer. The only difficulties can be parking and cell phone reception; even with validation it can be a little steep, and the building is infamously difficult to get a signal in.