Best way to travel from Rothenburg ob der Tauber to Erfurt

This is too much. You are going to spend more time traveling between towns than savoring and exploring Germany. The train is always best. I'd cut this way back if it were me - it takes MUCH LONGER than you think to check in and out of hotels and change cities. We love all the towns you mentioned, Nuremberg alone deserves more time.

I understand the focus of much of your trip, Munich through some key Reformation sites to Berlin. With that in mind and only 14 days, I would eliminate Rothenburg on this trip. We have been to all of the cities and sites you mention, some as recently as May of this year. All of the other cities/sites (except Fussen) fit in with your itinerary. Of the 4 east/west cities through Thüringen and Saxony, all could benefit from a close look. The exception is Eisenach where Wartburg Castle is a can’t miss stop on your trip. Maximize your time at the castle by taking a taxi up from the beautiful old train station and skip the heavily rebuilt city center. Call for a taxi or take the bus down. Pack light! You will be on and connecting on a lot of these former East German regional routes. You will need to store luggage in a locker at Eisenach. By eliminating Rothenburg and possibly Fussen you can spend a second night in stunning Dresden (my choice) or Leipzig. We spent the month of May this year to finally give some of these cities a closer look after dipping in and out as day trips on previous trips. These would be my suggestions. I think you’ll have a memorable glimpse rather than a hectic blurr.

We just got back from Germany. We spent 12 days in Germany. We spent 3 days in Munich, 4 nights. We toured the town first day, saw palaces the 2nd day and toured Dachau the 3rd day.
The train from Munich to Fussen takes 2 hours and 8 minutes with 1 change. I liked Fussen and we stayed 2 full days, toured town one day and went to Neuschwanstein the next day.
Went from Fussen to Rothenburg which was suppose to take 4 and 1/2 hours with 3 changes but took longer because of delays and missed connections.
From Rothenburg went to Frankfurt and flew home from there.
It takes more time then you think between cities. Often changing trains.
Just my recent experiences.
Robin

Welcome to the forum.

What's your planned arrival time at MUC? Even if it's early, I'd probably not think of Day 1 not as a sightseeing day. Consider possible flight delays, baggage + security waits, the 45-minute transfer into Munich, then the associated logistics of getting settled, and the potential for jet lag and fatigue from the transatlantic red-eye flight you just stepped off. That leaves 12 full days for seeing things in the 10 different destination cities/towns you mention. But those days are also full of transportation hours, which further subtract from your boots-on-the-ground time. Very little time is available for Munich. Your day trip to Füssen will require nearly 5 hours of transport if you're there for Neuschwanstein. The logistics of 7 different overnight spots subtracts quality time as well.

The streamlined itinerary below reduces the number of overnight stops and lengthens most of them. It also sticks largely to the main railway from Munich to Berlin. Some detours from the stops I name on this route, including Rothenburg, would be optional, so that you have more choices and fewer absolute deadlines for hopping on the next train. I've dropped Füssen and Dresden.

Night 1: Freising (super close to MUC, 11 min. by train; walk, eat, drink, relax in this comfy small town, adjust your inner clocks; try to stay up until at least 8 pm.)

Nights 2-5: Nuremberg

Nuremberg is a very fine choice and a great base for taking short outings.

Day 2: Freising > MUNICH (25 min's, direct early morning train.) Fully rested, you arrive in Munich, drop bags in a station locker, and carry out your sightseeing plans there. Munich > NUREMBERG (1.75 hrs, direct RE train in evening, or faster by ICE train.)

Day 3: 1 full day in Nuremberg

Days 4, 5: More time in Nuremberg? Or... outings by regional train to Rothenburg (1.25 hrs.) to the beautiful old-world art/wine town of Iphofen (45 min's) to Bad Windsheim's Franconian Open Air museum (55 min.) or to Würzburg ( 1 hr. 10 min's.) You don't have to decide now - these can all be done on the fly. Super-cheap and super-flexible day passes (VGN Tagesticket Plus) are available at ticket machines for these outings (but use the Bayern Ticket for Würzburg.)

Nights 6-8: Erfurt

Day 6 Nuremberg > (Bamberg) > ERFURT 1.3 hours by ICE train (or 3.75 hrs by regional train, ugh)

Stopover option en route to Erfurt: Bamberg, just 30-45 minutes out of Nuremberg, on the same railway line as Erfurt. It's a UNESCO World Heritage site for its medieval city center and worth at least half a day. Stow bags in a station locker while you visit.

Days 7 & 8 for Erfurt and Eisenach/Wartburg outing. You didn't visit Bamberg on Day 6? Maybe you'd find time for a 48-minute train outing through beautiful countryside to Mühlhausen.

Night 9: Wittenberg

Day 9: Wake up in Erfurt; catch morning train to LEIPZIG, drop bags in locker, full day for sightseeing. Continue by train to Lutherstadt Wittenberg in the pm.

Nights 10-13: Berlin

Day 10: Wake up in Lutherstadt Wittenberg (nice town to overnight in, IMHO) for a full day of sightseeing. Continue by train in pm to Berlin.

Days 11-13: in Berlin only

The daytime stopovers I suggested (Munich, Bamberg, Leipzig) might require specified stopover periods if you plan to use the ICE or other high-speed trains with saver fare tickets.

Wow, Russ (who is an expert on Germany and on rail travel) has done a great job for you.

Travel styles aside, I don't think you may fully understand how much (I mean, aside from your specific list of three objectives) you would miss, traveling at the death-march pace of your original proposal. For example, are you aware that Erfurt has a very important intact (if long ago deconsecrated) Synagogue? It also has a superb Soviet-era public garden (EGAPark), with its own tram stop. We happen to like public gardens, but this is the kind of place locals with strollers buy season tickets to, so it's a slice of "live-local", too.

Leipzig has a great art museum, and lots of history and literature. Frankly, the traces of Bach are a lot clearer than those of Martin Luther. I was also interested in Leipzig's role in the fall of the East German state. And don't forget the fictional tracks of Faust!

We visited the Andecks Monastery when we lived in Augsburg. The dark beer they serve there has 40% alcohol, so even after one beer, don't try to drive.

Thank you so much to each one of you who replied! Your advice has helped us thoughtfully reflect. We're in the process of making revisions by lengthening our trip to 15 days, not including the flights, 7 towns, and two night stays in each. We are active adults and think that the day trips will be exciting for us rather than tiring. We're not sure yet about Day 7 (3 leg journey) and Day 10 (3 leg journey). Those seem a bit strenuous. We'll study the train routes (time and costs) again tonight. Once again, we deeply appreciate each of your sharing your experiences and know-how!

Day 1: Munich Airport arrival by 9 a.m. > train to Fussen. (3 hour train) - gentle settling in that afternoon and evening.
Day 2: Fussen > hiking around Neuschwanstein > Munich in the evening (2 hour train)
Day 3: Munich
Day 4: Munich > day trip to Andechs Monastery (2 hour train round trip)
Day 5: Munich > Nuremberg (1-2 hours depending upon the train)
Day 6: Nuremberg
Day 7: Nuremberg > half day in Bamberg (one hour by train) > evening in Erfurt (2 hours total by train)
Day 8: Erfurt
Day 9: Erfurt > day trip to Wartburg Castle (2 hours transportation round trip)
Day 10: Erfurt > half day in Leipzig (one hour train ride) > evening train to Dresden (1-2 hours)
Day 11: Dresden
Day 12: Dresden > Berlin (2 hour train trip)
Day 13: Berlin
Day 14: Berlin
Day 15: Berlin > maybe a day trip Wittenberg? (2 hour round trip by train)
Day 16: Flight

Good revisions. You will appreciate your day trips to Wartburg Castle without the hassle of luggage and changing hotels. I hope you will do Wittenberg as a day trip from Berlin, great sites and home museum.

"Munich > day trip to Andechs Monastery (2 hour train round trip)"

There's no rail station at Andechs. The hour each way will put you on a bus for roughly half the trip. Since you describe yourselves as active and have hiking elsewhere in mind... an alternative for you might be the nice hike to Andechs after a 50-minute S-Bahn ride to Herrsching. Rick may have some hiking tips in his book. See link below as well. The hike will make you that much more appreciative of Andechs' refreshments. This is weather-dependent of course, so keep your train/bus schedule handy, or consider swapping days with the one set aside for Munich.

Trail advice:
https://wikitravel.org/en/Andechs

Speaking of hiking and "weather-dependent"... in the spring, hiking or any outdoor activities nearer the Alps, like in the Füssen area, can be more problematic yet. I would probably get together some indoor plans for the area if you keep your plan as is. The "castle" (Neuschwanstein is a palace) will only shelter you for the length of the tour, and that's a mere 25-30 minutes. That said, it wouldn't be my choice to make the 3-hr. train journey to Füssen immediately after a transatlantic flight in the first place - that's a tad arduous IMHO - and if it's only hiking that draws you there, it may be best to just avoid that one-night stay there on Day 1 as well as a booking that depends largely on weather conditions. Maybe Day 1 in/near Munich, then on Day 2, you make a day trip there IF THE WEATHER is cooperative. Munich of course has lots of indoor alternatives.

Can't really tell by your itinerary but I would suggest catching early trains to your next city ... not going to do much in the mornings except breakfast (grab a drink and some awesome baked goods or yogurt or whatever and have a breakfast picnic on the train ... get two seats across from each other with a table in the middle if possible) ... drop your bags at the next hotel and you'll have all afternoon in the city to start exploring before dinner.