I think Cleveland's art scene is quite impressive for its size, and I think surprising for most visitors and even for locals when they hear about all the art galleries and districts where you can find them.
University Circle is Cleveland's second downtown and one of the premier educational, cultural, and medical districts in the country. It is said that more institutions in these fields are concentrated within a square mile here than anywhere else in the world. This is definitely the place to start with the traditional museums. The Cleveland Museum of Art is phenomenal, highly underrated in the national scene, but is unfortunately closed due to a 250 million dollar expansion and renovation. Its collections are quite vast, and probably surpasses that of all other Midwest cities besides Chicago, and certainly better than any art museum you find in the South, Southwest, and West coast. You should still visit the district though. There is hardly a museum that doesn't exist here. Not to mention spectacular architecture like the Peter Lewis Building at Case Western, and Severance Hall - home to the Cleveland Orchestra. MOCA Cleveland is currently housed in the Cleveland Playhouse very near University Circle. The Cleveland Institute of Art in University Circle has good galleries to view.
Little Italy just east of University Circle is home to a lot of little galleries and neat Italian restaurants both old and new, particularly in a former school building on Murray Hill. In the summer they have a popular artwalk here. If you go further down Mayfield Rd from Little Italy you will hit Coventry Village. There aren't too many art galleries here, but definitely neat little shops and restaurants. Coventry serves as a kind of a sophisticated college town for the liberally inclined.
Downtown Cleveland doesn't have too much to offer in terms of galleries, but there are few. Tower City Center, the Cleveland landmark, transit hub, and downtown shopping center recently opened a great gallery called Lake Erie Artists. The Warehouse District along W.6th, W.9th, St. Clair, and Superior still has some remnants of the former art scene that used to occupy the neighborhood. But the scene there has become home to young professionals who live in these overpriced lofts and go to trendy bars and chic restaurants. The struggling artists who used to live in these lofts and had galleries have pretty much all moved from the Warehouse District to a neighborhood called Tremont. If you like art galleries and that scene than you have to go there. Its a neat neighborhood just over the River and South of I-90. It is a little tricky to get there, but very close. Find your way to Literary and Professor Avenues. Like Little Italy Tremont also has a popular artwalk in the summer. Tremont also has some of the best restaurants in Cleveland and the country according to national reviewers.
Just North of Tremont in Ohio City along W.25th is probably one of the most unique urban neighborhoods in the country. Working class people, artists, musicians, young professionals, and visitors from all over mingle in bars, cafes, and shops around the last city owned market in the country; the West Side Market. Visit on a Saturday afternoon when most of the shops and market are open and draw crowds. I like Something Different Gallery and shops like City Buddha, and Modern World. Just a little bit north of this strip on the Superior Viaduct is SPACES gallery which is said to be one of the most innovative galleries in Cleveland. The Superior Viaduct also has some of the best skyline views in the city.
If you are brave enough, you might want to consider visiting the real hole-in-the-wall galleries in the neighborhood just west of downtown along St. Clair and Superior. One example is the Tower Press building with Artefino gallery and café, and another would be the gallery in the former St. Josaphat church. This is a neighborhood in transition that was once industrial and a rough residential. A lot of Cleveland artists are moving in here, but it isn't quite developed into a Little Italy or Tremont yet.