2-3 week trip to Germany/Austria by train/bus

Frankly, even two to three months wouldn't be adequate for a laundry list like this . A good map ( google maps are a good bet ) is necessary , and start doing some practical planning . The trains are great , but the time it takes to relocate will generally kill half to a full day with each move . What you are suggesting is short on practicality An example or two - Weimar is well worth seeing and can be done as a day trip from Berlin , but are you familiar with the artists and architects who worked there ? Meissen is an interesting place . Are you a fan of the art and craft of European porcelain ? Gorlitz is wonderful but going there is about a two hour trip from Dresden . Dresden has great museums , but quite a distance from Vienna .etc. etc. You really need to narrow this down to a manageable format

Amazing! And kudos for taking yourself on this adventure!
I really like Rome2Rio for initially figuring out transportation as you are describing. I appreciate that you can get a sense of transfers and stops without having to navigate a train website. (If something looks hopeful then I use the actual train site.)
I only have experience with Munich- Salzburg by train. But an example of how ancillary time on either side can make the hours of a day shorter than you think.
I was meeting a RS tour in Salzburg and chose a train that was supposed to leave about 9 and arrive about noon. I planned some sightseeing for that afternoon before the tour meeting. Ha! There ended up being track repairs and the train stopped several stations prior to the main station- we were herded off and onto another train and by the time I actually got to the hotel it was almost mid-afternoon.
Happy planning!

In Rick’s Germany book he says to take the train from Baden-Baden to
Füssen, but this would take about 6 hours and a bunch of transfers. Is
Baden-Baden really worth it?

No. And Füssen/King Ludwig's Castles might not make your final cut either.

I think 7-8 days or so for Vienna and Salzburg (with an outing or two from there into the Alpine towns of Berchtesgaden / Hallstatt / Werfen (Hohenwerfen Castle) should adequately cover Austria and Alpine Bavaria.

After that... head to NUREMBERG and base yourself there for 6-7 days/nights; see N'berg itself but do outings from there to Rothenburg, to Bayreuth, to Bamberg (UNESCO World Heritage site), and perhaps also to Staffelstein (Vierzehnheiligen Basilika) and to Wuerzburg (Residenz.) Nuremberg is a perfect base town for seeing all these places by public transport.

Flying out of FRA makes sense; if you're really interested in castles and old-world half-timbered towns, then before you leave, I think you need to see Burg Eltz (Mosel River Valley) and Marksburg Castle (Rhine River Valley,) both of which are near Koblenz and the handsome towns of Cochem, Braubach, Bacharach, and Beilstein. Links to these and other area towns in this relatively compact area:

https://www.mosel-inside.de/en/travel-guide/mosel-villages.html
https://burg-eltz.de/en/homepage
https://www.mittelrheinentdecken.de/en/villages-towns-and-cities/braubach/
https://www.mittelrheinentdecken.de/en/villages-towns-and-cities/ruedesheim/
https://www.mittelrheinentdecken.de/en/villages-towns-and-cities/bacharach/
http://www.loreley-info.com/eng/rhein-rhine/castles.php

Trier, Germany's oldest city, is at the extreme upper end of the Mosel River in Germany. Rick Steves covers it fairly well in his book.

You'll probably enjoy a day cruise through the Rhine Valley as well (takes about 2 hours for Bingen > St. Goar.)

The Cologne Cathedral (Germany's #1 most-visited landmark) lies on the Rhine just to the north. In MAINZ, at the south end of the scenic Middle Rhine, the St Stephen's Cathedral and the Martinsdom (Romanesque) as well as the Gutenberg Museum would be worth your time as well. I'd spend my final day visiting this city and fly out of FRA the next morning (it's just 20 minutes from Mainz to the airport.)

5-7 days in this region might enable you to see most of these places.

Based on what you've said, I think that's how I'd pare down your choices.

Altho I did visit Baden-Baden on my first trip to Germany, I would skip it and the Black Forest ( even though I’m making another trip to the BF this year ! )

You really need to prioritize what you want to do and come back for more advice. Russ’s suggestions are good but may not meet your priorities. And everyone has different interests. I love Salzburg ( visiting 4 times ) and would go there in a minute. I talked it up to a friend who later went and she was totally unimpressed.

Of course Berlin, Hamburg and Frankfurt are worth going to. They each have very unique histories, and all 3 have so much to see and do.

As for the rest of your list, it is way, way too much for 2-3 weeks. Find 2 hubs and visit the towns that are near them.

I can help with Frankfurt and others can help with the other 2 cities.
Quaint towns near-by - Büdingen, Gelnhausen, Seligenstadt, Limburg, Marburg, Idstein and Kronberg.
Monastaries - Karmeliter Cloister in Frankfurt, Eberbach Monastery by Eltville on the Rhein and the monastery in Seligenstadt.
Top museums for Frankfurt - Städel, Judengasse, Historic, Film Museum
Interesting churches and cathedrals - Kaiserdom in Frankfurt, site of the elections for the Holy Roman Emperors since 1352 & the Justinus church in Frankfurt Höchst, possibly the oldest church in Germany, consecrated in 830. Deutsche Ordens church in Frankfurt, built in 1309 and with the oldest wall painting. St. Leonhards in Frankfurt, built in 1219 and a stop on the Camino.
Union church in Idstein, ceiling made of 36 oil paintings done in the style of Ruben, mid 1600s. Mainz Cathedral, and St. Stephens church in Mainz.
Frankfurt has a very long history of Jewish communities (1040s) including the 1st walled-in Medieval Jewish Ghetto and the 2nd largest medieval Jewish cemetery in Germany (1270-1828). Home of the Rothschild family, Paul Ehrlich and Anne Frank.

You guys are amazing! Thank you so very much! Steven- I know that my list is way too long to see in 2-3 weeks haha so I'm trying to narrow it down as best as possible. I don't want to spend my entire time on trains and buses! For Weimar, I'm very interested in German culture: Goethe, Schiller, and Franz Liszt (Hungarian but spoke German and lived here in Weimar). I was also interested in going to Bonn to see Beethoven's house and Leipzig for Bach. That town and castle in Werfen seems amazing; I've never even heard of it and I'm honestly shocked that Rick doesn't recommend it in his books.

I wish I could respond to each of you individually, but perhaps this forum doesn't allow for that. Rome2Rio is great- I just looked into it. Thank you BV! Vierzehnheiligen seems incredible. I saw it in a guidebook about a year ago and completely forgot about it. Thank you for reminding me. I love Baroque and Rococco architecture, and I was also looking at the Swabian Baroque Strasse a few days ago. I appreciate your itinerary Russ, thank you! I like your recommendation (and Jo's) of settling myself in one area (Nuremberg, Bavarian/Austrian Alps) and seeing sites in the area. Is this what is referred to as the "hub-and-spoke" method? I envisioned myself zipping all around the country at breakneck speed to see everything, but I prefer your method much more. Baden-Baden does seem slightly overrated to be honest, but then again there is a very strong Russian influence due to the nobility/modern-day Russian tourists, which seems extremely interesting, and Dostoevsky is strongly associated with Baden-Baden too which I like. As for Cologne, I would be interested in seeing the Dom as well as Museum Ludwig. I also want to see Potsdam with its palaces and the Berggruen Museum, as well as some others in Munich, Frankfurt and Vienna.

Salzburg seems amazing. I'm surprised your friend didn't like it, bigtyke. Think about the architecture/scenery/Mozart Kugeln! You guys are wonderful. Thank you again! I've heard Frankfurt is getting more and more dangerous, particularly around the Hauptbahnhof and surrounding area, but it does play such a strong role in Germany's history that I do feel obligated to visit the churches and museums, thank you for the recommendations, Jo.

I really like Rome2Rio for initially figuring out transportation as you are describing.

I would avoid Rome2Rio, as it is not that reliable and I have found errors there. Even if you use it just for planning, do not book anything on their website. Use the German National Rail website: https://int.bahn.de/en or if you just want to plan a route, Google Maps is a good option.

Sounds like you want to do The Romantic Road from Wurzburg to Fussen. Oberammergau is on the road along with Augsburg, Rothenburg ob der Tauber, Dinkelsbuhl and Donauwurth and Nordlingen.

I recommend this, but you need to rent a car.
https://www.romanticroadgermany.com

Berlin is well worth visiting. Hamburg is good, Frankfurt is OK. You have a lot on your plate for 2-3 weeks, but since you are doing Dresden, you could easily do Berlin.

Baden-Baden is in the Black Forest, and some distance from the rest of your plans. If you do it, visit Triberg, Friberg and Titisee.

I suggest that you get a map of Germany and plan your route to avoid backtracking. If you do three weeks, you should do one week in Austria, and that only leaves you with 14 days in Germany.

Sorry, but Frankfurt is more than just Ok. Yes, I live here and am passionate about the city, but honestly, it is so full of history, both old and new, that I fail to understand why it is just Ok. Enlighten me please.

"I was also interested in going to Bonn to see Beethoven's house"

I suggested the villages on the Rhine and Mosel Rivers partly for their location near FRA and for their half-timbered architecture and the abundance of castles, but also because of the sheer scenic beauty of rail travel through this area.

Mosel River scene: https://www.zughalt.de/wp-content/uploads/2011/10/DB11321.jpg

The scenery near Bonn is also very good. If "hub-and-spoke" travel appeals to you, this is a good place to pursue it. Bonn lies in between Cologne and Koblenz, the city where the Rhine and Mosel meet up. Find Koblenz, in the green "VRM" zone on the map of rail lines below, and think of it as a train hub (which it certainly is) and you'll see how it might serve as a solid hub for outings to Bonn and Cologne as well as to the Rhine and Mosel towns I've mentioned:

https://www.vrminfo.de/fileadmin/data_vrminfo/PDF/RLP-Ticket_streckennetz.pdf

Beethoven fans might want to visit this place in Koblenz, located at the base of the cliffs where Ehrenbreitstein Fortress towers over the area. (The cable car ride to the fortress is very popular as well.)

https://mutter-beethoven-haus.de/en/

I'm very interested in German culture: Goethe, Schiller, and Franz Liszt...

Maybe you already know this, but Goethe was a principle forerunner of the "Rhine Romanticism" movement in art and literature; Hölderlin, Brentano, Heine and others are known for their works which, along with British visitors to the area (Lord Byron and others) popularized the Rhine into a destination for traveling Germans and foreigners so many years ago. Liszt also has some Rhine-based compositions, as I remember.

Here are some museum resources for exploring the cultural history of this region:

https://www.romantischer-rhein.de/en/rhine-romanticism-and-the-region/monuments-to-bygone-ages-cultural-treasures/museums-on-the-romantic-rhine

In Koblenz: the Mittelrhein Museum:
https://www.mittelrheinentdecken.de/en/sights/museums/mittelrhein-museum-koblenz/

Not meaning to be pedantic , but You are aware that Liszt was Hungarian ?

@steven: B has already acknowledged Liszt's nationality in a previous post that names him as a person she's interested in:

"Franz Liszt (Hungarian but spoke German..."

I hadn't seen that , not having read the post that carefully

Absolutely Hamburg is VERY worth the visit. The Composer's Quarter might be of interest--Brahms and Mendelssohn were from here (and others), as would the Elphi (philharmonic). Hamburg tops the worldwide list in terms of tall churches (we are the only city to have five over 100m), each with great history and with samples of the UNESCO intangible heritage recognition of organ craftsmanship. See the Schleswig-Holstein post for quaint villages near Hamburg.

But maybe not on this trip give the list you already have! I would also strike Görlitz just from a geographic standpoint. It's cool, especially when combined with places like Bad Muskau / Rakotzbrücke, but not worth losing the time you would need getting there that could be used for other, better places. I would also strike Bonn unless combined with the Rhein region in general.

Another thought comes to mind . As a number of these destinations are distant of each other , you will eventually need to make additional trips . Here is one example to keep in mind - You mention Leipzig and the connection to Bach . As you will likely make it to Vienna on this trip , one of the numerous places to visit might be The Seccesion Building . In 1902 . the Secession show was a tribute to Beethoven crowned by Klimt's Beethoven Frieze and centered around the centerpiece of the event , the magnificent statue of Beethoven by Max Klinger (not the character from MASH ) The statue itself is not in Vienna , but in Leipzig . so eventually another voyage might take you there , and it is not to be missed . Also , a concert by The Leipzig Gewandhaus Orchestra would be an experience as well https://en.wikipedia.org/wiki/Beethoven_(sculpture)

B,

Sounds fantastic. Please let me add the following ideas. First, as you start to plan your specific locations and their respective sequence, use Google Maps as if driving to get an idea as to distance and direction. Second, will you be using German's D-Ticket? This is a great deal as it covers local travel on trains, trams, and busses in Germany. You will need to buy ICE train tickets for longer trips. Since you are heading to Austria, the D-Ticket is good all the way to Salzburg if you originate in Germany.

Have a great time. And, as your plan develops with more details you may want to add a new second or third post as this one will get hard to follow and we many not be sure what you have decided.