Five weeks or so in Germany, summer 2019

Sounds like a nice time overall.

"I do want to keep the trip enjoyable, and not a whirlwind either.. "

Even with 5 weeks, you are covering a lot of ground. I suspect you will need to whittle down your very long list of destinations.

"Next area of focus, Thuringia, with possibly a home base in Erfurt, which seems perfectly situated to see a lot with easy day trips. Ideas are Eisenach/Wartburg castle, Weimar and Gotha, maybe some random beautiful town like Muhlhausen."

Before you buy the Länderticket, you should investigate the other local options as well. Ticket machines in Erfurt, for example, sell normal tickets for shorter trips that are priced according to travel distance, as well as the VMT "Hopper-Ticket." (VMT refers to the local transit zone - see map at link.) For €9, the VMT Hopper-Ticket provides a round-trip for one to destinations 50 km or less from the base town. Erfurt-Weimar-Jena West - Erfurt is one example of such a trip (see the #4 blue route on map at link above.) Erfurt-Arnstadt-Erfurt would take you outside the VMT zone; for such trips of 50km or less, the Thüringen-wide Hopper-Ticket is available.

There is also the Regio 120 ticket for certain trips; normally you can find this option at the DB website, along with the Länderticket option, when you do a search.

"...with us staying in modest budget type accommodations..."

We found a nice apartment with the right location and at an outstanding price in 2016 using the Thüringen-Tourismus site for information and for booking as well:

https://buchen.thueringen-tourismus.de/ttgs/ukv/?lang=en/

"Nuremberg and Frankfurt are probably the two base towns with the most options for things to see and the most diverse attractions which can be reached from there."

Nuremberg indeed is a convenient base town for outings as long as you're staying in a Nuremberg location near the main station. Nuremberg lies within the VGN transit zone (which offers the VGN Tagesticket Plus, a cheaper alternative for train outings than the Bayern Ticket.) Large cities like Nuremberg and Frankfurt are not necessarily the best base towns for everyone, however; smaller towns, if you prefer those, can work just fine too if carefully chosen with your outing destinations in mind. We are "base-town" travelers as well and have found that we like a nice mix of large and small. In Thüringen we made our base in a small, quiet farm town of less than 3,000 residents for several days. It was great to take in the expansive valley outside the large living-room window in the mornings, then swing it open for the pleasant farm aromas; back "home" after our train outings, we'd stretch our legs a bit in the evening air along the footpaths that led to the river nearby.

... perhaps one of either Frankfurt/Wiesbaden and other Hessen places
Recently I had a pleasant week in what is called Middle Hessen, with Wetzlar and Fulda as a base. From Wetzlar wonderful places like Limburg (a/d Lahn), Weilburg and Marburg are easily reached. Fulda has the nearby Fasanerie (a 20-minute ride on a half-hourly bus) and low-key Rothenburg (not to be confused with that other, apparently irresistible Rothenburg).
... plus special trip to Oldenburg
Northern Germany is much underrated. Have a look at cities like Lübeck (one of my favourites) and Wismar. The suggested Rostock wouldn't be high on my list, having suffered from war and post-war damage.

Here is a map of Germanys highlights according to Baedeker, Germanys most popular travel guide:
http://imgur.com/9BcKenL

In my view you miss one important part of the country: the Baltic coast. There are basically three different types of towns in Germany: towns made of timber, stone, and brick. The towns of Northern Germany are made of brick, and they look completely different.... darker and more medieval. The Baltic coast is moreover Germanys most popular summer destination, mostly because of the sandy beaches, but also because the region is a great mix of nature and culture.
So, I would recommend to travel from north to south, or vice versa. This way you see everything. Brick in the north, towns full of timber-framed houses in Central Germany, and towns made of stone in the south or in Saxony.

My list in eastern and North Germany to choose from culturally and historically contain both cities and small towns: Leipzig,

Meissen, Greifswald, Weimar, Minden an der Weser, Jena, Eutin/Holstein, Wustrau/Brandenburg (on Prussian history), Neustrelitz,

Celle, Lüneburg, Kiel, Naumburg an der Saale, Münster/Westf, Soest/Westf., Frankfurt an der Oder, Schwerin. Cuxhaven, Lübeck-

Travemünde, Erfurt, Magdeburg, Hameln, This list includes those I only visited once and those repeatedly, plus a few not as yet.

Thanks everyone for your ideas on this.
@Lubitsch, I like the idea of varying what we see, not going from one similar place to the next, also the idea of home base towns which have a diverse set of attractions, such as those you mentioned, Nurnberg and Frankfurt. Such places are ideal for what I'm looking for.
@Russ, thanks for the link to the VGN site, it has a great list of suggested places to see reachable through public transportation in the area. I spent two summers in the Nurnberg area when I was a teenager, but missed seeing most of the attractions.
@Tonfromleiden and Martin, I am torn about northern Germany and the Baltic coast region, about whether it might make sense to keep it for another trip or incorporate it to this one and cut something else. Also thanks for link to the Badeker map, it provides a nice overview of things.
@Fred, I love your lists of towns, some of which I've heard of others not, but they usually look interesting when I research them.
For now I think I'm going to keep up the research, but keep the overall outline of my travel style intact, but I can see the destinations might change. I think my wife and I would enjoy such a travel pace the most.

Hi,

Logistically, you can do both North Germany and the Baltic coast, depending on your final choice in the itinerary, say Hamburg, Schwerin, Lüneburg, Lübeck, Kiel , Plön, Ludwigslust (the Schloss there), even Cuxhaven. For example, Hamburg to Schwerin...direct shot by train, then to Rostock. Study the main train lines, most have Hamburg Hbf as the junction point.

"...focus on Potsdam and other palaces." You have several to choose from. Potsdam is not merely seeing Sans Soucci as it relates to Schlösser. There are numerous Potsdamer Schlösser...Neues Palais, Babelsberg, etc.