End of July is Browns training camp in Cleveland and it should be electric this year! My sister and I hit up Cedar Point for the same reason. I would probably aim for a mid-week day for coasters. Kings Island is a sister park and I know last year you could get a pretty good deal for both of them.
July is a great time in Ohio. Weather is awesome and there are festivals everywhere.
https://ohiofestivals.net/ohio-festivals/Put-in-bay is a fun area to take a free out to near Sandusky. More adult party atmosphere. Kelley's Island is similar but slower and family camping atmosphere. Great views of the lake. Cuyahoga Valley National Park (southeast of Cleveland) is gorgeous as is Hocking Hills State Park (east of Columbus).
If into baseball both the Reds and Indians have fantastic parks with some very loyal fans.
Cleveland - Christmas Story house is a fun tour, Rock & Roll HOF is fun too. Westside Market is also enjoyable. Avoid Football HOF, it suuuuucks.
Columbus is very much a food town. It has one of the best zoos in the nation (though those don't seem of interest). Also have quite a few family friendly barcade locations which are lots of fun for a relaxing time. I'm in Columbus and frequent the museums but they aren't geared for teenagers.
Cincy has an amazing art museum that is free and the park it is in is up on a hill overlooking the Ohio river.
Dayton has the Air Force Museum which is fantastic.
Mansfield had the Ohio State Reformatory which I've heard great things about. Popular with ghost tourism.