Without much notice, the Cleveland Museum of Art abruptly closed a great portion of its exhibits a few months ago, and word is they'll be gone for years. Even in such a discombobulated state, this is still the finest repository of art you'll find in the city.
There are a few dozen galleries still open, and you can spend a good afternoon wandering around. The most beautiful art is all in one hallway right now; they've put out their very favorites with little regard for chronology (O'Keefe, Renoir, Picasso, and Pollock), and it's beautiful.
The staff is friendly and courteous (they have a nice way of reminding you that, no, you can't photograph copyrighted pictures - whoops!). I came in khakis and a T-shirt and only felt a little underdressed.
Easter Egg: the courtyard has a sculpture of a lion done by the same fellow who did the stonework at Squaw Rock, another unmissable local outdoor attraction. Look for his signature.
The fact that it's free only makes this a better museum. It's always quiet, cozy, and full of treasures.