Munich and Prague

Logistically the best would be to fly to Munich and back to US from Prague so called open jaw. If it is much cheaper to fly first to Frankfurt and connect by train to Munich you can do that. I would leave Berlin for another day. I would leave my days for Munich and surroundings and the same for Prague. The fastest connection between Munich and Prague (besides flying) is Deutsche Bahn double decker bus. It's nonstop, leaves from Munich Main Train Station and goes to Prague Main Train Station. For schedule and prices see Deutsche Bahn website. For big cities like Munich and Prague you don't need a car.

With two weeks, you can certainly see Munich, Prague, and Berlin. You won't want or need a car for the cities themselves, but a car could be useful for destinations outside the cities (particularly in Bavaria). If you are planning to drive from Germany to Prague, be sure your rental car can be taken in the Czech Republic (there are often restrictions on this). To investigate train times and routes, use the Bahn website, following Rick's tutorial: https://www.ricksteves.com/travel-tips/transportation/trains/online-schedules. To look at estimated driving times and find routes, use Via Michelin: http://www.viamichelin.com/ Even though it's cheaper to fly round trip to Frankfurt, don't forget to account for the time and money it will take to get from FRA to your intended destinations (although there is a train station in the airport itself). And if you're ending in Berlin, look at the prices to fly back from there.

Have you found Munich to be that much more expensive to fly to than Frankfurt? I never have. But, assuming you do want to fly to Frankfurt and then go to Munich, you have several options. One option is to take a train directly to Munich from the airport, but full fare tickets are fairly expensive, 98€/p for a 3½ hr ICE. If you book advance purchase Savings Fare tickets (as low as 29€/p on the same ICEs), these are train specific and non-refundable, so, with a late flight, you stand the chance of missing your train and having to buy new tickets. As an alternative, you could go first to somewhere nearby, such as Bacharach or St. Goar and using the non-refundable ticket to Munich a few days later when you know you can make the specific train. You could also book the non-refundable ticket for a few hours later, when you could reasonably expect to make the train, but in that case, you might as well take the first regional train connection to Munich and arrive about the same time. You could do this with local (RMV) tickets from the airport to Kahl am Rhein, the first stop in Bavaria (7,60€/adult), and a 26€ Bayern-Ticket (also purchased at the airport) from there to Munich (you don't change trains in Kahl, just tickets). The RMV ticket would cover your travel by S-Bahn to the Frankfurt Hauptbahnhof and by Regional Express (one every hour) from there to Würzburg, Nürnberg, then to Munich. Or, since you will probably want to see Rothenburg during your trip anyway, go there first, since it's less time on the train than all the way to Munich.

I would fly to Frankfurt and visit Nurnberg, Munich, Bamberg, Regensburg (and maybe down to Salzburg) using rail system and then take bus to Prague (I think that's the best way - so I've heard). I would spend 10 days Germany/Austria and 3 or 4 days in Prague. I would skip Berlin - but that's just me (and I've never been there so what do I know?

I realize you live in a far different location than do I, but we usually use United or Lufthansa. There is very little penalty for Open Jaw on those two, and we have always gotten one connection (I mean, like Newark-Berlin-Cologne, arriving by noon the next day, and Amsterdam-Newark for $100.00 more at most. Your mileage may differ. We presume that you have queried your airline website repeatedly, and tried the "Flexible dates" option, as well as "Multiple Stops". But I have put in trips like "EWR-CGN", and the site has created the connecting itinerary for me.

If you do include Berlin (at the end), there are 6 direct Eurocity (EC) connections from Prague to Berlin per day that take under 5 hours, and they have a Europa-Spezial fare as low as 29€/P. The Bahn bus is certainly the fastest way (4¾ - 5 hrs) from Munich to Prague, and, if you purchase tickets early enough, at 29€/P, sometimes the cheapest way. However, you need to purchase the tickets well in advance (up to 92 days) to get that low fare, and the tickets are bus specific and non-refundable. There are three direct trains from Munich to Prague per day and they only take an hour longer. You can use a Bayern-Böhmen-Ticket for 2 people, purchasable from an automat anytime for 29,50€ (2€ more from a counter for personal service) to go from Munich to Pilsen, and a Czech Rail ticket from Pilsen to Prague, for 100 Kc (~4€). I'm told you should be able to purchase the Czech tickets from the conductor on the train in the Czech Republic. He might take Euro, but it would probably be a good idea to have some Koruna (purchasable from WF at about $6/100Kc) with you.

"a car could be useful for destinations outside the cities (particularly in Bavaria)" Germany, in particular Bavaria, has one of the most extensive rail systems in Europe, augmented by regional buses, such as the RVO (Regionalverkehr Oberbayern). In over 2 month traveling in Bavaria, I have never found anything worthwhile I couldn't get to easily by public transportation. If you can get there by car, you can probably get there by bus. Use the German Rail (Bahn) website to find schedules.

We loved Prague, but I don't think it's worth a week by itself. Some other attractions are close by, like Karlovy Vary, and Marienbad. But Cesky Krumlov was very important to me, and as Rick reports, it's faster to drive than to take the train for a day trip. There are other Southern sites too, like the "Budweiser" brewery and the "Three T" towns, very historic. So think about a car in Czech Republic. If you haven't been to Budapest, another choice is four nights in each of the three, and no car fuss.

Try to fly directly into Munich and out of Berlin. Use Munich to Prague Bus. Use Prague to Berlin train: EC trains are non-stop with no changes involved,are 4:45 hours.See website www.bahn.de or www.bahn.com Berlin is very doable with two weeks especially if you do it at the end. Trains from Berlin to Prague run later within the day and would use more of your time.
Trains from Prague to Berlin leave throughout the day and offer more time advantage options:(629 am,10029am, 129pm, 1429pm, 1629pm). From your three bases there are endless public transportation options to do side trips. Listen to Lee as he has a lot of experience.

Let me make Diane's information more accurate. There are no nonstop trains between Prague and Berlin. Deutsche Bahn is showing six stops between these two cities.

I think Diane means direct (i.e., no changes) rather than non-stop. The only non-stop trains I know of are the rare ICE Sprinters, which run, for example from Berlin (Spandau) to Frankfurt Hbf in 3h24m, non-stop. For one person, the Munich-Prague bus starts at 29€ with an advance purchase, non-refundable, bus specific Savings Fare ticket. A single person Bayern-Böhmen ticket, purchasable just before travel, and a Czech Rail ticket will cost about 29€. One hour longer trip vs advance purchase, non-refundable ticket. For two people, the Munich-Prague bus starts at 58€, while a Bayern-Böhmen ticket with Czech Rail tickets cost about 37,50€. For three people it's 87€ (min) for the bus, 46€ for the train. The bus does have a snack bar.