Niall,
Cleveland has a ton of great golf, more on the private side, but some wonderful public as mentioned. I just finished 18 at Big Met, a Stanley Thompson with some good holes and only $23! Sleepy Hollow GC is better, with very few weak holes and many great holes. Bring your A putting game and stay be low the hole! I have not played everything yet, but have played the large majority of courses in the last three years since moving back to Northern Ohio.
My favorites:
1. Country (The Country Club)
2. Kirtland
3. Brookside (Not in Cleveland, but on a half hour away)
4. Hawthorn Valley (a Ross gem that has blind shots and need many trees taken down)
5. Aurora Golf Club (Bert Way who did Firestone South Orig. and Mayfield)
The last two would not make most peoples top five, but I love old corky courses. They need tons of trees to fall, the pictures from when the courses first opened are amazing.
Jonathan, we should get out and play, along with Evan.
Hope Dean D. sees this and can add because he has played everywhere.
Chris