I'm from Holmes County (Amish Country) originally. Berlin especially but also Sugarcreek will be the most packed and touristy especially in the summer and fall. There are also plans for road construction (not sure when exactly) so things might get worse.
The main strip of Millersburg following 39 has some fun shops and a brewery.
It also tends to be less busy than the east side. I would personally skip New Philly/Dover as it is a bit out of the way.
Mt. Hope has a huge yard sale/auction the last week of June. Also try Mrs. Yoder's Kitchen there. I prefer it to Farmstead in Berlin.
You can also try Cincy or Cleveland for a baseball game. Cedar Point if you're into thrills. In winter, Clifton Mill is worth it to see the lights come on. If you're into airplanes, the Air Force museum in Dayton is fantastic–in some ways better than the Smithsonian.
I've pondered more than once about going to Utica to the Velvet Ice Cream factory.
Cuyahoga Valley National Park (Peninsula, OH) has some sections of the canal system and a working lock.
There was a place where the canal once passed over the Cuyahoga River is pretty fascinating accomplishment,even by todays standards. Very scenic.
Velvet Ice Cream in Utica (on the way to/from Millersburg and Amish country). The murals on the floodwalls in Portsmouth and the mounds in Chillicothe and Newark. Fort Ancient on the way to Cincinatti has interesting history, too.
Fairport Harbor, east of Cleveland, has a great beach for swimming and boating. There are two lighthouses in town (one out in the water and another in town). The one in town has a marine history museum attached and you can walk up in the tower for a spectacular view of the area. Some very good food at a restaurant along the lake at the east end of town.
Tour some of the President's homes (Marion, Mentor, to name a few), the Neil Armstrong museum is very good. Football Hall of fame in Canton, too..
Clifton Gorge, near Yellow Springs is worth a hike, and of course Hocking Hills (plenty of cabin rentals in the region).
The Wilds, out past Zanesville, is supposed to have some really exotic wildlife out in nature (they give driving tours through the park). Zoos in Columbus, Cleveland, Cincinnati and Toledo.
There is much history in every town if you ask around. Check out weird things and unusual history. There were many inventions and innovations that started in Ohio, so explore. Pick a direction and stop at the roadside historic markers!
Pick a theme like covered bridges!
There are many still accessible. Just outside Ohio: The Henry Ford Museum in Dearborn, Falling Water (Frank Lloyd Wright's most famous house) in Ohiopyle, PA. Pittsburgh also has some great architecture and museums.