Probably depends where in Cleveland you'll be. Clevelanders refer to themselves as Eastsiders or Westsiders, and the they are pretty geographically separate.
My brother used to go to The Brew Kettle (http://www.thebrewkettle.com/bop/homebrewing.htm
), located in Strongsville (West Side), but I read somewhere that they might be closing.
Another place that's very well-regarded is the Grape and Granary (www.grapeandgranary.com
) in Cuyahoga Falls, a bit of a trip south on I-271 and OH-8, if you are starting on the East Side. That's where my bro goes now.