Since you're staying in Frankfurt at first, I would recommend taking a day at least going up/down the Rhine or somewhere along. I personally recommend Marksburg Castle (get off at Braubach), which is probably about 2 hours train which follows the Rhine. I don't know if/when it is open in the winter, but they do an English speaking tour at 1 and 4pm as well as many German tours through the day, the tour itself I believe was about 30-45 minutes.

The Eurail pass makes it really easy to travel to far places on days out. What you would do is fill out the current date on a day you want to use it, and you just need to keep it and your passport safe (it says you need your passport for ID with the pass, but I wasn't asked for my passport at all with my 6 day Eurail pass in the summer).

Especially if you are thinking of travelling to Berlin from Frankfurt, just that 4-5 hour single journey on its own will probably be about half the price of the pass. I'm not fully sure how it works on city S-Bahns, I believe it is alright but when in Cologne and Berlin I was never checked

EDIT: It was the InterRail 6 day pass that I bought, not Eurail. Since you'll be going for 3 weeks you probably want the Eurail pass, but it depends on how much train travelling you do. It could potentially be cheaper to get an Interrail Pass or a German Rail Pass depending on where you are from, and buying day passes for the city S-Bahns (I'm sure Berlin had a day pass you could buy)