A week somewhere after Berlin

If you post again, please give the month of the year and describe the family's travel history a bit.

I'm inclined to suggest Cologne or Duesseldorf. The latter has the second-largest airport in Germany. Cologne has excellent train connections, and is good for several days on its own. Places in Cologne like the Applied Arts Museum (MAK), the Ethnographic, and the Medieval Art museums, and the Roman museum might appeal nicely to younger guests.

Berlin is easily good for a week, even without daytrips. And it has superb rail connections.

What about Salzburg as your base? Good centre for day trips into Germany and the Salztkammergut if you want more mountains. Lots to do in the city itself. Good train service to Munich, Vienna etc.

Opinions will be all over the place. Every one of us has favourites and biases. So here is my recommendation without knowing your boys' ages or more about your likes and dislikes:

Hamburg, of course! In addition to the normal sites, here are things locals are more likely to do or things that are more kid-friendly:
--We have beaches right in the city. With shipwrecks. The Blankenese Treppenviertel is really pretty, too.
--Great parks and playgrounds--an epic one right in Planten un Blomen (park), which also has beautiful flowers and in the evening a cool water fountain / light show.
--You can use your normal day train ticket to ride the ferries, so for younger kids, it's "let's go for a boat ride." There are official harbour tours, of course, but these are more expensive and more informational. But you can get close to the really big ships if that is of interest.
--We have pirate lore. The Hamburg Museum has pirate skulls. I mean, if you are stuck in a museum because of crappy weather, what kid doesn't love skulls? It's a super popular field trip destination--kids love that museum.
--Miniatur Wunderland.
--Sand dunes for walking/nature lovers and people interested in weird geology. Very pretty. Options in Holm ( or Boberg (in the city).
--Hagenbeck Zoo was the first in the world to have natural looking animal enclosures instead of cages. It's a really cool place, and if you go early you can feed many of the animals, including the elephants.
--Kiekeberg open air museum lets you see how life in northern Germany was in earlier times, has animals, a fantastic water playground, and is generally very popular for kids. Nearby Wildpark Schwarze Berge is also neat.

Depending on when you are traveling, the possibility of the Hamburger Dom (big fair, free entry with paid rides and games)

Fantastic day trips:
--St. Peter-Ording to the Wattenmeer (Wadden Sea) beaches. If you can (depending on ages, etc), ride the horses. It's a UNESCO site.
--Lüneburg / Lüneburger Heide (heath)
--Schwerin
--Haithabu Viking Village, a UNESCO site
--We have a very unusual animal sanctuary for endangered farm animals. Lots of feeding of weird goats that look like Tim the Enchanter from Monty Python. It's called Arche Warder
--Towns like Glückstadt, Stade, or Jork that are pretty, though perhaps not as interesting for kids. Although chasing sheep along the dike in a place like Glückstadt, while technically frowned upon, can be fun for smaller kids.
--Helgoland, especially if you can find a place for a one night stay rather than a day trip, though a day trip is neat. See seals up close--perhaps closer than you would like!
--speaking of seals, maybe the Friedrichskoog Seal Station, where seals are rehabbed and, if possible, released
--Bremen

Most of these are free or not expensive options, and they are what locals do.

From what you describe you should probably get your family to somewhere in the Alps.

The Salzkammergut would be a good choice. There's an absolute bumper crop of family fun vacation stuff to do there, some interesting towns, and you have Salzburg, a beautiful interesting city. And also Berchtesgaden, a beautiful German national park that was the Nazi mountain headquarters, is right there.

You would have lakes with beaches with dedicated kid fun zones, lifts, hiking trails, mountain coasters, mine tours, Alpine huts with sun-drenched patios, bike trails, big beautiful playgrounds, ice cream shops, pastry shops, some interesting Habsburg history, cowbells, Hills alive with the sound of etc etc. There are a lot of places to rent for a week. Red Bull is headquartered there so I suppose you can get about as gnarly as you want to :-)

Only slight downside given your criteria is that the Salzkammergut tends to be on the more expensive side for Austria's Alps. This is not to say that it's expensive compared to Switzerland or Northern Italy -Austria is a relative bargain. But for Austria it's a bit pricier than some other places in the Austrian Alps. But I would say worth it. Which is not to say you couldn't go to Tannheimtal or Schruns cheaper and have just as much mountain fun. It's just that the Salzkammergut isn't lacking anything.

There's a bunch of other places I could suggest, but help yourself narrow the field by insisting that you stay somewhere close to a pleasant body of water (if summer time). A lake or the sea with proper vacation town amenities is going to give you a built-in satisfying thing to do with your kids on a regular basis. You're not going to be bored If you have some activity you go do most days, and then you're able to return to your home base for a swim and an Aperol Spritz well the kids play on the diving boards and floating inflatable fun zones.

Another tip, after Iceland, Denmark and Northern Germany I would hedge my weather bets by heading south. The weather doesn't have to be nice in any of the places you're going even in the height of summer, and then you could end up spending a cold beach week up there too. Conversely anywhere from the foothills of the Alps south are a much better bet for fine weather.

And it's just as easy to fly to Iceland from Munich et al. as it is from Northern Germany or Denmark or wherever. It's just 20 minutes longer from Munich than from Berlin for instance.

I'm going to second Howling Mad's recommendation for Hamburg. You can get just about anyplace from there, and there are lots of wonderful day trip to seaside places. And as HM said, there are a ton of kid-friendly things to do there. I know one family who went there just so their kids could go to Miniatur Wunderland.

It would definitely be cheaper than Salzburg or Bavaria, and at the end of your trip, you're much closer to Copenhagen and Iceland.

I saw from another post of yours that you will be traveling in the summer. In my opinion, that's a good reason to stay farther north. :)

We are going to be spending a week in the Harz Mountains later this year. Might be someplace for you to look into -- lots of hiking, a narrow gauge train system, quaint towns.

I would stay up north too, Hamburg, Potsdam, Quedlinburg, etc. Add in the beaches and the water, and your kids will love you.

https://abnb.me/DLxHUF83qyb

Quite reasonable for a full week in July in excellent central village.

My suggestions include a number of those towns already recommended above.

First of all, stay in North Germany to visit and explore Lüneburg, Hamburg, Eutin/Holstein, Schleswig, Flensburg, Schwerin, Husem, Kiel, the Kiel Canal, Minden an der Weser, Hildesheim, Goslar, etc.

Or, head into eastern Germany...numerous lovely captivating places associated with German cultural history...Weimar, Naumburg an der Saale, Halle, Leipzig, Erfurt, Magdeburg , Rheinsberg, the greater Berlin area and Potsdam, plus the boat cruise on the Havel, Babelsberg, Neustrelitz, Stralsund,

If you opt for a rental car, I recommend going out into the rural areas of Brandenburg, ie, out in the sticks to see the villages in an area of the former Mark Brandenburg, the core Prussian area, among others. Or, do a driving tour with the rental car, going from Berlin to the Oder.

part 2 here....Seeing the landscape from Berlin to the Oder, either laterally to Frankfurt an der Oder or a bit more southerly to Küstrin-Kietz I've done these routes by train, obviously, and a few times motorised....very enlightening, and sociologically and historically revealing. Plus, this geography one recognises only that of belonging in Brandenburg.

Thanks everyone for your input. It was super helpful. Though I've been to Salzburg, I haven't been to the surrounding area, and didn't know about Salzkammergut area. Sounds like our kind of place, so I've booked a week there in Mondsee. i think there is so much to do in that area that we will have plenty to keep us busy in the outdoors. Not really as inexpensive as I was hoping, but "oh well", you only live once. :)

Woodgiesubs, I've ridden a bike through Mondsee and the inner town is lovely. If you stay there and use public transit, you'll end up passing through St. Gilgen basically every day trip you take. St. Gilgen (or Strobl, or other surrounding places) are better tourist towns closer (or walking, or ferry ride) to a lot more you'll want to do. They are fully in the mountains while Mondsee is on the edge of the mountains. The train runs direct to Mondsee, but the buses in the Salzkammergut are as good as the train - big, clean, fast, frequent, efficient and easy to use.

Mondsee will be great - nothing at all wrong with it. Somewhere on the Wolfgangsee would likely be optimal.

But again, not a big deal, honing an already sharp knife :) Have a great time.

@Hank, thanks for your input. The reason I chose Mondsee was it looked like it had stuff going on without being as touristy as St Wolfgang. I also booked a place originally right in Fuschl but was worried it would be a bit too "isolated" for us. We like to be able to walk to town and restaurants, etc. and not rely on a car or public transport. and thought Mondsee would give us that. But, I also found a nice place right on the water in St. Wolfgang. In your opinion would that be a better area to be? Is there one town that is better for being accessible to bus/public transport than another? I'm struggling to find much information about train/bus lines in the area, and thought we would just need to rely on a car for day trips.
advice much appreciated!!

I'd suggest starting a new thread to zero in on what works for you in the Salzkammergut - lots of experience on this forum.

Also take a look at this topo map with color coded elevation. Really helps me visualize how each lake is situated relative to the mountains.

https://en-gb.topographic-map.com/map-cvtgt/Europe/?zoom=11&center=47.77641%2C13.36633&base=2

Yes St. Wolfgang is touristy. It's also a bit of a disbursed area, so if you are a couple hundred meters out of the center it's not so touristy feeling. Personally I'd pick St. Gilgen on Wolfgangsee, and then probably Strobl, but I'd also be perfectly happy in St. Wolfgang. It feel slightly more impersonal as a tourist hub but the upside is a lot of amenities. I've stayed several other places in the Salzkammergut as well, Gosausee, Hallstattersee, Bad Ischl. The first two are great but kind of cul de sacs you'd need to get out of when sight seeing, and later also great but no lake which makes it not as good for paddle boarding ;)~.

The place I haven't stayed in the Salkammergut but would is Ebensee am Traunsee. I've tried to book there a few times and couldn't find a place, but on paper (in pixels?) it has everything I want - traditional village, lake with all the lake stuff, surrounded by high hut-speckled mountains, lift network right out of the town, still pretty central to stuff you'll want to go do. The lake is a bit bigger, looks a bit more like Lake Como in its geography, ferry to Gmunden which is a bigger pretty city on the other end. It hits me as a place that German speaking families rent for a week more than Jane and Jim from Ohio passing through for two nights on pilgrimage to gaze upon Hallstatt.

Re transit I've not used a car in The Salzkammergut and haven't felt the need. But it could be fun to drive to more isolated spots too.