Hi Leah,
I'll start off with a couple of negatives: I'm yet another guy (strike 1) and I'm currently, er, between Rivs (strike 2). But I have lived in central Ohio since 1995 (man, where did the time go?), so I'll give you my $0.02.
I'll plug my area: Dublin is a great place to bike. It's an affluent suburb NW of Columbus. It has more than 150 miles of multi use paths and bikeways, and they actually connect places. For instance, we now live in Jerome Village, a sprawling development just to the northwest of Dublin, and I can ride from our house and be downtown to the library and shopping in a pleasant nine miles. And since this is Ohio, the bike paths aren't very heavily travelled for the most part, so it's not like you're on a congested freeway dodging speeding e-bikers.
The Bridge Park area in downtown Dublin is also an interesting new development: they've made a mixed use urban center in the middle of a suburb, and it really works. Shopping, dining, activities, housing, businesses, riverfront, all in a short stretch. And I can bike to it easily.
To get a little feel for Dublin, I'll toot my own horn and suggest you visit my blog, which I started when the kids were small and I decided to take them to every park in the city (56 at that point!) by bike:
Spoiler: we mostly succeeded. Dublin even gave us T-shirts afterwards.
There's also some decent county roads north of us, but I mostly stick to town these days. I feel like too much of a target on a two lane road. But there are group rides that leave from Glacier Ridge metro park and head that way. Maybe a women's ride, I'm not sure. I think these are run by Columbus Outdoor Pursuits.
Otherwise around Columbus, Clintonville is a great neighborhood that has a nice bike connection to campus and downtown, but that trail is heavily used, and not all that great to ride on. I'm guessing you may also look at Upper Arlington or New Albany (no biking feedback here, but I don't like them because they always beat my daughter's team in sports) or Lewis Center/Powell, which I would avoid due to heavy sprawl.
Feel free to pm me if you any more specific questions.
Eric