Hi, Cathy. Welcome to the forum and congratulations on going to Germany! You're going to love it. You'll find a lot of helpful people here on the forum (and a grump or two who have good info to share but sometimes in rather blunt ways).
Germany has many similar-named towns/cities, so you have to be watch your spelling when doing research (and buying train tickets!). Since you mentioned the Black Forest, I assume you want to go to Freiburg (in Breisgau), a large university city on the western edge of the Black Forest, not Freiberg (am Neckar), a small city just north of Stuttgart and somewhat northeast of the Black Forest but closer to Nürnberg/Nuremberg. The difference in endings exists because, in German, "berg" = mountain and "burg" = castle.
I don't have a lot of familiarity with the Frankfurt area, including the Rhine and the Mosel Rivers, so I will leave it to others to offer suggestions there. I think most of Day 5 will be lost to travel, by the time you get to the train station, catch the train, arrive in Nuremberg, leave the train station, find your hotel, and get settled. That might make a trip to Bamberg difficult to fit into your schedule. Berlin is awesome -- a little bit of an outlier to the north relative to the rest of your itinerary, but doable.
In the Black Forest, there is a card called the KONUS Card that allows for free travel on trains/buses in the Black Forest, but you have to lodge/stay in certain towns. Sadly, Freiburg is not one of them. The KONUS Card English flyer currently linked on the Black Forest tourism website is here and includes a list of towns where it is offered. The Baden-Würtemmberg-Ticket is a regional day ticket that allows for discounted travel all day on Saturday/Sunday and after 9 am on weekdays for travel within Baden-Würtemmberg (which includes all of the Black Forest) on "local and regional services." One person travels for 24 euro (all day) in 2nd class, and each additional passenger travels for 6 euro up to 4 additional travelers. I would think that plus point-to-point tickets for your other trips would likely be cheaper than a Rail Pass.