Downtown Attractions:
Tower City
Public Square
E.4th St.
Arcade
Cleveland Public Library
Civic Center
Rock and Roll Hall of Fame
St. John the Evangelist Cathedral
Playhouse Square
Jacobs Field
Warehouse District
I have a recommended walking tour to highlight downtown's sights, and most attractive streets with good architecture and restaurants. You should start early and give yourself about half the day.
I would start with visiting Tower City. It is a beautiful large indoor complex of shops and restaurants in the city's landmark skyscraper. It is also the transit hub of the city and anchors Public Square in the heart of downtown. After Tower City consider stopping in the historic Renaissance Hotel Lobby (attached to Tower City) for a drink or just to see. You will want to go outside to Public Square, and then I recommend going down East from Public Square down Euclid Avenue. When you get to East Fourth Street the Arcade will be on the left. After Tower City this is the second most important building to see. It was the country's first shopping mall and really beautiful. You can through the Arcade to Superior Avenue. Once you are on Superior cross the street and stop in the Public Library. Make your way around the Public Library to the Civic Center mall. Walk north all the way down the civic center mall and take a right on Lakeside. When you hit E.9th take a left and you will be at the Rock and Roll Hall of Fame. The Rock Hall is the most popular attraction downtown Cleveland and you should definately go.
If it is a Saturday you are visiting, you really should go to the West Side Market. It is only open during the day Friday, Saturday, Monday, and Wednesday. Saturdays are the best time to go. Hopefully your downtown walking tour started early and your visit to the Rock Hall finished up around 2PM. Outside the Rock Hall you will see a rapid transit station with RTA marked on it. Buy an all-day pass for $3. Board the Rapid back and get off at Tower City. Transfer to the Red Line and board the Rapid marked "Airport". The first stop will Ohio City W.25th St. Outside the transit station you will see the West Side Market across the street. Go in, check it out, try some food, enjoy the atmosphere, etc. It is awesome and only a few markets like it exist in the U.S. It is almost exactly like a typical European big-city market. After the West Side Market check out Cleveland's famous micro-brew Great Lakes Brewing Company across W.25th Street along Market Avenue. After that you could stay in the neighborhood and eat dinner somewhere because there are a lot of great restaurants, but I would recommend heading back to Tower City on the Rapid and walking to the Warehouse District on W.6th Street where the nicest and most highly trafficed restaurants and nightlife establishments are located.
If you are not visiting on a saturday or a day the market is open, then after the Rock Hall go back down E.9th. This is the city's financial district. St. John's will be on the left, and you might want to stop in. It is a neat cathedral. Keep going down E.9th until you get to Euclid and take a left. When you hit E.13th you will be in Playhouse Square. When E.13th crosses Euclid it becomes Huron, and I recommend heading down Huron to make your way back to Tower City. Huron goes through the heart of the Gateway Neighborhood which has a number of casual restaurants. sports bars, and the Q Arena and Jacobs Field. Take a left on E.6th and that will put you in the center of the two sports venues. Go back up E.6th (or Ontario if you happen to be there after walking around the stadium) and take a left on Huron (you are essentially back at Tower City). Take a right on the narrow W.2nd Street where the Hard Rock Cafe is. Take a left on Prospect. Propsect becomes W.6th St. where the city's nicest restaurants, nightlife establishments, and residences are in the Warehouse District. In May it will be especially nice to visit because many of the restaurants and bars have sidewalk patios.
University Circle Attractions:
-All the city's museums except for the Science Center and Rock Hall
-Wade Lagoon
-Little Italy
You should be able to do downtown and Ohio City in one day. If there is another day with time to kill you might want to consider checking out the city's second hub called University Circle. This is a really beautiful area with a lot of museums circling a lagoon and park. The art museum is closed, but there are a number of other museums. If the museums don't interest you at least check out the lagoon and take pictures there. Cleveland's Little Italy is just east of University Circle at the intersection of Mayfield and Murray Hill. In my opinion it is a great Little Italy and better than New York's. Here you can grab some great pizza, gelato, or other food.
The Cleveland Music Festival will be going on Mid-May at pretty much all the downtown music venues. Maybe this is why you are coming in town, but if you haven't heard about it you might want to consider dropping in at one of the venues to catch some bands. It is all rock music and it is kind of big deal. It is going on May 11-13.
Nightlife and other considerations:
In the evening there are a number of places to go for people with different tastes. The Warehouse District is where most of the nightlife and upscale dining is. There are casual and inexpensive places here, but not very many. If you prefer to "do something" other than eat and drink in the evenings then go to E.4th St where you can see some music at the House of Blues or comedy at Pickwick and Frolic's. There is also a decent Irish pub called Flannery's on E.4th St. If you prefer more chill neighborhood bars and restaurants than stick to Ohio City on W.25th Street. If you want to go where all the artists or creative people in Cleveland go, then head to Tremont (which is south of Ohio City, but you can't walk there from Ohio City). If you are a politically active liberal and don't eat meat, then you would most fit in at Coventry Village. Lots of funky restaurants, shops, and bars along Coventry Road at the intersection of Mayfield. Coventry is located just East of Little Italy (but you can't walk there from Little Italy). A good family thing to do in the evening would be to go to Shaker Square for dinner and a movie. Gay bars and clubs are on the West Side and spread out along Detroit Avenue between W.25th and W.65th, and then there are some gay bars a few streets north of Detroit on Clifton east of W.117th.