Take a daytrip out of Rome to Tivoli. It’s got three of the best things:
Villa D’Este an old palace with a hughe terraced garden with the most beautiful fountains and a great view of the Lazio countryside.
Villa D’Adriana an old estate with tonnes of ruins that you can walk around at your leisure, I was there 3 hours!
Villa Gregoriana This is a main reason I am going to Tivoli when I go back this summer. Google it. Really, it looks amazing. 100 ft waterfall anyone?

I’m not really into visiting tons of art musuems either, and went more for the culture side of things. I really liked Siena because it was old and charming. there’s a really great bike trail called the Po River trail that runs from outside Venice to Ferrara. Ferrara was a nice bike-friendly place where you can rent bikes. I was on bike, so I’m not sure how you would get there, but there were a lot of hiking trails in the Apennines. I went through Parma to Berceto and down to La Spezia.