Greetings from the state that is round on the ends and high in the middle :) Yup Ohio :) I can help with the western side of the state - I haven't been to Cleveland much, so I am no help there. I live in Columbus and went to college down near Cincinnati and half my family is in Toledo.
Jungle Jims - It is a huge international grocery store. It is truly unique. Not only can you get food from all over the world there, there are hidden surprises at every turn. In the British section, you can see Robin Hood, but if you look up in the middle of an aisle, you will see his gold stash. They have a great beer and wine selection. The cheese section is out of this world. https://junglejims.com/ - Go to the original one in Fairfield.
The American Sign Museum is a place on my to visit list the next time we go down for our German fix :)
It is also fun to be along the river. You can cross the river on the Purple People Bridge. and go over to Newport KY. (They have one of the 6 official Hofbrau Hauses in the US outside of Munich)
Speaking of German, Cincinnati has a huge German population with active German clubs. If you hit the time of the year right, there could be something going on. Oktoborfest and the Christmas Markets are popular. The coolest Christmas Market is at Germania the weekend before Thanksgiving. The others start after Thanksgiving and are downtown Cincy. http://germaniasociety.com/
My favorite place to go is for eating is at Tony Packos. The Original on Front St is the best. The one at the Ball Park would be my second choice. The others aren't as fun. It is a Hungarian restaurant known for it's hot dogs. So if you remember MAS*H this place was mentioned several times by Jamie Farr's character (who is a Toledo Native). The fun thing about this place, is they have famous people sign hot dog buns. https://www.tonypacko.com/.
Toledo also has a great Art Museum and Zoo.
In Maumee which is town outside Toledo - there are two historic places. First is a marker for the Battle of Fallen Timbers (1794). The second is Fort Meigs, which was part of the War of 1812. We were at a British Car Show at the Fort and they were filming for the History Channel on the War of 1812 so that was cool. Also Maumee is a river and if you are into fishing you can fish there for fish that normally live in Lake Erie that come up the river to spawn.
The National Museum of the US Air Force in Dayton is worth visiting. Also if you hit it right, there is a Air Show sometime in the Summer. Not sure when. As a kid I remember it being really fun! The Wright Brothers were born not too far away, so there is a huge Aviation presence in Dayton. https://www.daytoncvb.com/things-to-do/aviation-sites/
Not too far north of Dayton in Wapakoneta is the Armstrong Air and Space which is also on my list of places to visit
I am going to respectfully disagree with the poster above about skipping Columbus. Yes, things are a bit crazy around here during football season, but Columbus is more than just The Ohio State University. I will say it is an interesting campus to walk around. (But Miami University down near Cincinnati in Oxford is better)
German Village and the Book Loft are fun places to visit. The Book Loft has over 30 rooms of books. It is a fun maze to get lost in. German Village also has a German Restaurant called Schmidt's. Which has been on the Food Network. It is famous for it's sausage and cream puffs. When Arnold Schwarzenegger is in town for the Arnold Classic you might catch a glimpse of him here. German Village has an old world feel to it. It is fun just to walk around.
Speaking of Arnold, there is a statue of him downtown at the Convention Center. Every March people from all over come to compete in Health and Sports for the Arnold Classic. He brought it to Columbus as he won his first body building competition here.