The road portion of the bike way is not the in the greatest condition but easily rideable. You can ride the trail up to Martin Luther King Boulevard and follow the Harrison Dillard trail
http://planning.city.cleveland.oh.us...HDbikeway.html The surrounding neighborhoods aren't the greatest though. If you ride the trail east it goes to Bratenahl , which is a very nice section of beautiful lakefront homes. I wouldn't go any further than Bratenahl though.
If you are comfortable riding on the roads you can ride from spin and into downtown and easily get your 20 miles if you ride the lakefront trail. The ride would not be very touristy from Lakewood to downtown.
Ohio city is also a neat place to ride, it's just outside of downtown. There is the west side market
http://www.westsidemarket.org/about.html The Great Lakes Brewing Co
http://www.greatlakesbrewing.com/ and a few other resturants and bars. I am probably giving you way too much info...lol