Amsterdam, Germany, Prague 16 days

All in the western part of Germany can easily be reached from Amsterdam. You apparently not being the kind to hang around a lot, why not a roundtrip starting in Hamburg, for big-city life, or nearby Lübeck, if you prefer it more historical? Have a few days in Stuttgart, with cities like Karlsruhe and Heidelberg around the corner.
The best place to get to Prague is Leipzig or Dresden. Dresden is very touristy, and Leipzig is amazingly attractive. From both places it's just a few hours to Prague.
In between you can have a stopover in Bamberg (very pretty) or Nürnberg (with its WWII associations).
Everything can easily be reached by frequent train services. Planning is best done on the website of the German railways (www.bahn.com/en/view/index.shtml). Long-distance trains come with a huge discount when bought well in advance (90 to 120 days) if you don't mind being stuck to a specific train (day and hour).

"Would like to use train travel as much as possible but would rent a car in Germany if necessary."

There's no way a car would be necessary to have a nice 10-day trip in Germany.

"...would like to limit the places we stay to 3 or 4 and day trip."

Day tripping requires some extra travel time so you want to pick base towns that are train hubs and have numerous options for outings. Try to stick close to your A'dam - Stuttgart - Prague route in the interests of time. Some suggestions, all doable by train:

After A'dam: I'd suggest BOPPARD in the Middle Rhine Valley. Not a train hub but a well-located town for outings by train to the many castles and villages on the Mosel and Rhine Rivers. A river cruise from Bingen back to Boppard makes for a scenic outing as well.

STUTTGART is a good base. Tübingen, Ludwigsburg, Esslingen, Calw and many other destinations are within train-outing distance.

NUREMBERG is an outstanding base town for trips to Bamberg, to Würzburg, Iphofen, Rothenburg, and Bad Windsheim.

And Nuremberg is a gateway city for trips to Prague.

What would be a good itinerary in Germany?: I would say - coming from Amsterdam: Hamburg, Rhine and Mosel area - scenery, castles, wine; Black Forest, Bavaria, Berlin.
Jump off point to Prague?: Dresden - it's only two hours by train from Prague. It's on the way from Berlin to Prague. Or Nuremberg -just three hours from Prague nonstop by Deutsche Bahn run bus.
You can find a load of information on this website. At the left hand side of this page click on Explore Europe then click on the country you are interested in then region or city and also go for every link there including you tubes and recommended guides and books. Also suggestions of itineraries for each country and city listed there.
Once I spent the time and went through all Rick Steves' Europe website and I could say that I found the answer to almost every question asked on Travel Forum on this website. And also one can go to this Travel Forum and look under the country - a lot of questions already answered there.

Hi,

On Stuttgart: If you are into German lit, there is the Nationalmuseum of Schiller in the small town of Marbach am Neckar, reachable by the S-Bahn from Stuttgart Hbf. If it's history you want, see the Königsplatz

Amsterdam 3 days
Germany 10 days - perhaps Cologne (2) > Rhine (Boppard) (3) > Stuttgart (2) > Munich or Nuremberg (3)
Prague 3 days

Choose Munich if you are considering days to the Bavarian Alps or Salzburg.
Choose Nuremberg if you are attracted to days out to Regensberg and Bamberg.