Best order to visit these cities?

Where are you flying into and out of, or is that not yet decided? That will dictate the start and end points.
Then string them together in the order that will give the shortest distances. Look at a map.
Bern is second division, spend the time in the Alps. Then you have to decide where in the Alps; Switzerland, Austria, Italy, France.
The real big question, why this particular choice of places? There are lots of other places you could insert in the circuit. What sort of places are you looking to see. Big cities, museums, old buildings, history, little towns, mountains. The choice is unlimited.
3 weeks = 21 days = 20 nights = 5 places x 4 nights, or 6 places x 3-4 nights.

We are looking to see mountains, small towns and would like to stay somewhat off the beaten path. Suggestions other than these cities? Thank you!

We've traveled out of Amsterdam up the Rhine River 3 times over the years by auto--ending up in Munich. We stopped at Cologne and Bacharach each time going up river. And two times we went down into Tirol and the Innsbruck area to get our fill of The Alps. We find it's very easy to make a circle through the mountains south of Munich--ending up in Salzburg. Switzerland is a little out of the way and much more expensive to travel through than the mountain region of Austria.

The website rome2rio.com is good for checking various options. In general, if you do trains, it's cheaper if you book them in advance (ideally about 90 days before). Value airlines are an option for some of the longer distance jumps, but it's keep in mind the time of dealing with the airport and possible costs in getting to/from the airport.

Amy, you wrote "We are looking to see mountains, small towns and would like to stay somewhat off the beaten path. Suggestions other than these cities?". In which case you do not want cities. But, with two boys you probably want a lot of activity, so a river cruise is out. How old are the boys, that will affect your choice.
Is there a reason you are starting (and possibly ending) in Amsterdam, or is that also to be decided? I am trying to find out what the limitations are on your planning. You won't get mountains (or hills of any kind) in the Netherlands, it is famous for being flat.
Yes to some time in the Alps, Austria or Italy will be cheaper than Switzerland if you are on a budget. You need to look up some small towns in Germany, along the Rhine Gorge, or elsewhere with town walls and castles; I assume this will appeal.

• If you start in Amsterdam, take a day trip to Delft.
• If you can stretch to a car for one day, take a trip to the Delta Works, this is mega-engineering on a scale that should appeal to boys of all ages.
• As above, stay in a town in the Rhine Gorge.
• Somewhere in Bavaria. Rothenburg ob der Tauber is famous, but there are many more equally as good.
• At some point in your trip, be near a mountain where you can take a cable car to the top. Mountain resorts have lots of activities.
• If you can end up, and fly home from Italy, that would avoid a double-back north, and give a good contrast between northern and southern Europe. Depending on their age, Venice may be a good place to end up.

Sorry that is a bit vague, but you are covering a large area.

I would travel ...
Amsterdam >> Berlin >> Bern >> Alps >> Munich >> Vienna

Amsterdam - Bern - the alps , Munich, Vienna then Berlin.

Berlin is pretty far out of the way. You will save time by making only one long trek to Berlin, rather than one to Berlin then another long trek back to Vienna.

It's about the same time Amsterdam > Bern as it is Berlin > Bern.
Vienna > Berlin is considerably longer, however, than Amsterdam > Berlin.