Here are a few ideas, if the links don't register live cut and paste.
The Wilds, near Zanesville
www.thewilds.org/ ;
Columbus zoo, (national #1 rating) <cite class="_Pd" style="color: rgb(0, 102, 33); font-style: normal; font-size: small; font-family: arial, sans-serif; line-height: 16px; white-space: nowrap;">
www.colszoo.org/</cite>;
Columbus Art Festival
June 6-8 (280 artists flyer says nationally acclaimed)
http://columbusartsfestival.org/
Tecumseh outdoor drama!,
Murals in Stubenville;
Hocking hills in Southeastern Ohio;
Hocking Valley scenic train rides
http://www.hvsry.org/
Cuyahoga national park
http://www.nps.gov/cuva/index.htm
Athens Dairy barn (cultural arts center for Athens) currently has art on paper display, free on Thurs evening. On odd calendar years -2015 next they have Quilt National display June-Sept (labor day) if you will be here next summer.
www.dairybarn.org
And Rock and Roll Hall of Fame is in Cleveland.
Astabula County has at least 17 covered bridges, including a modern largest one in Astabula itself.
If you are anywhere near Oberlin Lorenzo's Pizza family owned and wonderful
http://www.lorenzospizza.com/
We found this ice cream chocolate family owned shop on the way back from Cleveland Airport it was a unique shop.
Malley's Chocolates
<dl style="margin-right: 0px; margin-bottom: 15px; margin-left: 0px; width: 230px; color: rgb(0, 0, 0); font-family: arial, helvetica, clean, sans-serif; font-size: 13px; line-height: 16.00299835205078px;"><dd class="adr" style="margin: 8px 0px; padding: 0px; color: rgb(51, 51, 51); font-size: 16px;">23473 Lorain RdNorth Olmsted OH 44070</dd><dd class="tel" id="phone-11641970" style="margin: 0px 0px 0.6em; padding: 0px; font-weight: bold; font-size: 26px;">(440) 777-4500</dd><dt class="websiteLink" style="margin: 0px; padding: 7px 0px; display: inline; font-size: 14px;">Visit:</dt> <dd class="url" style="margin: 0px; padding: 7px 0px; display: inline; font-size: 14px;">
malleys.com
</dd></dl>