Berlin to Prague

The best idea is to take one of the direct trains from Berlin Hbf to Praha hl.n (travel time about 4.5 hours, Saver Fares currently listed at €39 PP). IMO, I wouldn't bother with "places to see/stay/what to do in between". There's lots to see and do in Prague and with day trips from there to places like Cesky Krumlov.

Definitely take the train, don't rent a car or anything as Berlin to Prague is well served by train (plus huge drop fees for picking up in one country and dropping in another). You could look at Dresden or Goerlitz (filming location for the movie The Grand Budapest Hotel) as places in between, or even Karlovy Vary which is a spa town. I have not been to any of these places except Berlin and Prague, but have had my eye on the places in between for a number of years. You can check train routes at bahn.de.

There are lots of things to see and do in Berlin. Get a good guide book and decide how much time you want there, including a day trip or two (Potsdam comes to mind immediately). Dresden is a good stop by train, about 1/2 way between them. Much of the city's center been rebuilt, stone by stone and is very pretty. There are a also a few fine museums.

On rereading your post, I am unclear about what you are asking. You wrote: we need to be in Prague by Oct 21

Do you mean that's when you want to start sightseeing in Prague? Or that's when you are leaving Prague and want to spend time there before the 21st? Or that you don't plan to spend any time visiting Prague? It's only about 4 hours by train from Berlin to Prague. Or do you want to stop in small towns and villages?

Dresden is between Berlin and Prague. Even with a train specific, discounted, advance purchase ticket from the Bahn, you can use the stopover function to spend 24 hours, or so, in Dresden on the way to Prague.

Dresden deserves much more than just one or two days, especially when some people here spend up to four nights in a tiny place like Rothenburg ob der Tauber. That's just absurd. Dresden is probably the best base for day trips in Germany after Munich.

Dresden itself:
World-class museums (Green Vault, Old Masters, Armoury, Turkish Chamber, Porcelain Collection etc.), reconstructed old town with Zwinger, Frauenkirche, Semper Opera (do at least a guided tour), and the more lively Neustadt area. Very nice are the suburbs (Radebeul, Wachwitz, Loschwitz), historically wine villages which were turned into posh mansion districts.

Day trips:
Preserved towns: Meissen (like Prague in small), Görlitz (best preserved German town), Bautzen (Baroque toen surrounded by medieval fortifications), Freiberg (miners town), Pirna
Castles and fortresses: Pillnitz, Königstein, Moritzburg, Weesenstein
Nature: Saxon/Bohemain Switzerland, Elbe valley
Transport: historic paddle steamers on the Elbe, nostalgic steam train in Radebeul, funicular and suspension railway in Loschwitz
Shopping: wines (Radebeul), porcelain (Meissen), Christmas (Seiffen/Ore Mountains, Germanys Christmas country)

Thank you, everyone, for the fantastic advice. Yes, Chani, I realized I was a bit unclear about what I meant by we need to be in Prague by Oct 21. I meant that we have reserved a hotel in Prague that begins Oct 21. We spend four nights in Prague and we fly back to Canada on Oct 25. We are looking for something to do/see/where to stay in between leaving Berlin on Oct 13/14 and our hotel reservation in Prague on Oct 21. Karlovy Vary sounds awesome.. What exactly is a "spa" town? And does anyone know anything about the Festival of Lights in Berlin? We will be there during that time.

Also in between Berlin and Prague is one of my favorite cities in Germany - Leipzig. There are two wonderful churches in the city - the Nicolai church and St Thomas. The Nicolai is a several hundred year old church with a great recent history. It was where the famous Monday night candle light marches which eventually brought down the Berlin wall began. The St Thomas church was Bach's church. He was the organist and choirmaster there for decades. Additionally Leipzig is the home of Auerbach's Cellar, the restaurant in which Goethe set one of the best scenes from Faust. You can have an excellent meal there while looking at the walls painted with scenarios from Faust.