Not living in Columbus myself just yet, but I'm going to be moving there from Cincinnati, which is roughly 2 hours south from Columbus in southern Ohio, so I've got some cred at least lol.
For hiking and such, definitely look towards Hocking Hills State Park! It's about an hour south from Columbus. The nice thing about Columbus is that it's about dead center in the state, so you're not too far away from really anywhere, so make sure to look at places that aren't just in Columbus! Look around Cincinnati, Toledo, Cleveland, and more. There's tons of places. My personal favorite hiking spot in Ohio is Buzzardroost Rock! It's roughly 2 hours south from Columbus, absolutely gorgeous view at the end. Some other places I like include:
Germantown Metro Park
Caesar Creek State Park
Glen Helen Nature Preserve
John Bryan State Park
There's also some great places in Indiana such as Versailles State Park, which is a further drive. Red River Gorge in located in Kentucky and is a fantastic area, but it is about 4 hours away so that's probably reserved for breaks when you have the time lol. One of my former coworkers got me a book for Secret Santa one year that is all about hiking spots in Ohio, and I've really liked it! Definitely going to be using it myself at tOSU since I'll be closer to more northern spots I never felt like making the trip to go see! It's called "50 Hikes in Ohio" by Ralph Ramey.
My biggest tip for living in cold weather is LAYERS! Being able to layer on clothing is a massive help, especially when you are dressing for the temperature outside and inside. Getting a nice winter jacket/coat that you can layer along with something like a sweatshirt or pullover is really nice and gives you control over how warm you are inside buildings too, as you can adjust how many layers you have on just in case it's hot in one room and cold in another and freezing outside. Cover your extremities, such as hands and ears, when it's really cold out, so gloves and hats are great! Scarves are nice too to help cover your face. I love jackets that cover my neck due to personal style preference, but they also have great use in keeping my neck warm in the winter months.I personally like wearing long socks in general, but they're also very nice for the winter months so my calves aren't as cold. I am a guy who definitely runs on the warmer side typically, so I don't always follow all this advice myself, but for someone who is coming from a much warmer climate who isn't acclimated to a colder climate, I feel that it's at least a good starting point!
Also, I just worked a double shift today, so I apologize if any of this seems disjointed or all over the place lol, I'm a little tired.