Some wise advice please!

For the distances you plan to travel the train is the best option. The train from Berlin to Prague takes less than 5 hours. A flight probably more, because you have to include the waiting time at the airport for check in, security check up, etc. Normally the trainstations are in the center of the towns/cities, while you need sometimes more than an hour from the center of town to the nearest airport. Taking the train would also give you the possibility for a stopover in Dresden.

It is the best route. I totally agree with Liberat. Moreover there are cheap train tickets available 90 days before travel time. With regard to your travel date (Thursday off-season) it is very likely to get tickets for 29 Euro or at least below 50 Euro. If you want to do the stopover in Dresden you can choose this in the advanced options at www.bahn.co.uk It is possible to add a stopover lasting up to 48 hours.
There are also bus offers like www.berlinlinienbus.de but the train trip is more comfortable.
As far as I know the express bus from Prague to Nuremberg has to be booked in before. I think it can be booked at www.bahn.co.uk as well.
If you don't want to book online it's also possible to book at the machines in Berlin main station but the cheap budget tickets are normally sold out two days before travel time and you might have to pay the standard fare...

The main stations in Prague and Nuremberg are both close to the city center. If you choose your hostel/hotel carefully it will be in walking distance to the train station and city center.

You'll have one full day in each destination--that is, one day when you'll wake up and go to bed in the same place. With bits and pieces of the days before and after.

These are ridiculously short amounts of time to spend in major European cities. Most experienced travelers would find the ratio of time spent getting to places vs. the time spent in them, well, unacceptable.

Heaven forbid you should actually like+ one or more of these cities. But I don't think you'll have enough time +anywhere to make that sort of determination.