Scandinavian trip.

Just a few partial answers:

I use Google Flights to search schedules and airplanes for transatlantic flights, then I book directly with the airline. Don't be tempted by third parties like CheapOair.

Tell us where you're flying from and others may have suggestions for budget airlines that might work for you, like Icelandair or Norse Atlantic Airways.

Your destination countries are very expensive (especially Norway), so it will be important to have solid information on your travel options. Rick's guide to Scandinavia will be helpful. He suggests penny-pinching techniques that you may want to take advantage of from time to time. (As for me, I bought a lot of cheese, nuts, bananas and Wasa crispbread at supermarkets in Norway and Sweden.)

If you fly into or out of Oslo, the regular train from the airport will be your cheapest option. It also allows you to use city transit from the train station to the area of your hotel if you need to. I think that's good for 90 or 120 minutes after you buy the train ticket, which you can do right at the airport.

I don't know the best website to use for hostels; you can probably find that easily by Googling. I've notice that booking.com lists some hostels, so you might also look there. Do be careful about locations. One way hostels keep their rates low is by (often) being located in less-than-central parts of town. You are planning to move very fast, and it would be best not to be staying a 45-minute bus ride from the center of a city.

You can do the Norway-in-a-Nutshell trip--at least in the Oslo-Bergen direction--without spending a night en route, but it will be a long day. I spent a couple of nights at the hostel in Flam so I could see the Borgund stave church. I had a private room with my own en suite bathroom. I imagine they have a room with three beds in it. Furnishings are simple, but the place is clean and well-located. Like everything else in Flam, it books up very early. If you want to break the NiN trip midway (edited to add: I meant without spending extra time in the area; if for more than one night, it's worth checking out Balestrand), Flam is the best place to do it, and you should book a room at the hostel ASAP.

Train tickets between major cities and ferry tickets are both likely to be cheaper if booked early. Be very careful about cancellation/change policies.

Others will be more helpful about your routing, but I will point out that Bergen has good airline service, so flying between that city and Denmark or Sweden (in either direction ) could be efficient.

While I understand all too well the cost of traveling in Scandinavia, I think it's a real pity to have only one day in Copenhagen, Oslo and Stockholm. An extra day would be nice in Bergen as well. I'd urge you to try to add a few days to your itinerary so you can have a better look at the major cities, especially Stockholm and Copenhagen.

I would like to stay 2 nights in each city of Oslo, Stockholm,
Copenhagen.

That's a very short stay in those cities, if you can add a bit of more time you will not regret it.

I already know I want to bike the cities--we are all bicycle riders,
or do a Hop On Hop Off bus/ boat and realize with just two nights we
will just get a general idea of each city.

All cities are easy to explore by bike, although parts of Stockholm and Oslo are a bit hilly. The public transportation in all cities is great though and metro, buses, trams and ferries are also good options.

Perhaps one museum and one thing the kids might like.

What do they like? For Stockholm, the Vasa museum and Skansen can be a good combination for a day.

1. best priced flights? Into Copenhagen? Stockholm?

Who knows? Predicting flight costs is tricky and your guess is as good as mine. But Copenhagen and Stockholm are the largest airports in Scandinavia and are well connected. Between the US and Scandinavia, SAS, www.flysas.com is often a good option. They have direct flights from Chicago to both Stockholm and Copenhagen.

Best places to stay there for four? Close to airport/station.

Stay in the cities, do not stay around the airports.

1. Best route? Copenhagen to Aalborg to Bergen to Oslo to Stockholm to Halmstad to Copenhagen and home?

That's not a bad option, although you can also do it anticlockwise or start in Stockholm.

How do i get from Aalborg, Denmark to Bergen?

Train to Hirtshals and then ferry to Bergen.

Should I do Sweden first, then Oslo and Noway in Nutshell and Bergen,
then to Aalborg and back to Copenhagen?

Maybe, there is really no right or wrong answer to that question.

1. Anyone have a estimated cost for flights? rail? most hotels and hostels?

Trains should be booked in advance for longer trips, they can get very expensive if you try to buy tickets last minute. For hostels in Sweden, STF is a good start, https://www.swedishtouristassociation.com. A memorable place to stay in Stockholm can be Af Chapman, https://www.swedishtouristassociation.com/facilities/stf-stockholm-af-chapman-skeppsholmen-hostel/

1. Who can help custom this trip and advise on travel insurance?

This forum is not a bad option.

Playing Devil's Advocate here. I wonder if you view this trip primarily as a "heritage" journey or as a quick over view of some parts of Scandinavia? I ask because I've done both and felt that they were very different trips. The heritage was more in-depth while the other was more touring and seeing out the window while on transport.

To me, and I am not you - we all travel differently, this is a lot in only two weeks. Have you tried laying things out day by day? There are more travel days and short stays than I would enjoy, but that is me. If you used a guided tour like RS's as a model, remember that they have buses at some points for quick, specific travel routes. Public transit may be less efficient. Not that it can't be done, but may make for a lot of coordination, timetable checking, and focus on departures and stations rather than enjoying the current location.

I wonder specifically if you might consider dropping Norway and/or Norway in a Nutshell for this trip? I am probably out of the norm, but my travel companion and I both found the Oslo-Bergen train quite boring. We traveled early July this summer, your time frame. Everything we read promoted the route which was easy, but to us it was long and, frankly, dull after the first several hours. We didn't do the extensions to the fjord or Flam train. Perhaps that would have made the trip worthwhile, but we didn't want to do an overnight (as suggested above) due to time and other travel priorities, and we decided we didn't want a day of 10 - 12 hour travel between Oslo and Bergen. I'm just giving you another perspective as it is one I didn't find in any of my research, and frankly, our reaction to the train journey surprised us. I had been looking forward to it and was disappointed. You may well feel quite differently!

My reason for asking about the above is that if you wanted to explore the ancestral areas, that would give you more time, and as suggested, you could add time to Copenhagen and Stockholm stays. Two nights equals one full sightseeing day and, at best, partial days on arrival and departure.

If you wanted a "taste" of Norway, Bergen is quite a fine stop from where you can also take fjord cruises, hike and overlook the city and fjord, and see the historic wharf. Other options available like composer Grieg's house, art museum, etc. Take a look at the cruise options by Rodne. We took a quite enjoyable 3.5 hour cruise and there were also full day options. Hurtigen also has cruise options. The fjords are unlikely to be as dramatic as on the full NIN route, but we were both quite satisfied. Multiple, easy plane connections to Copenhagen when we flew. If you do NIN, book early. The route was sold out when we traveled last summer. I understand that is common.

If you indicate your grandchildren's interests on this thread or a new one, I suspect you will get good ideas for activities and sites. I have lots of thoughts out of Copenhagen and Oslo but will tailor them later.

One more thought, I - personally - would not ride a bike in Copenhagen, but that is me. Too much bike traffic for me. It is a very walkable city - flat and sights are close.

Edit - I forgot to say how much I admire you for wanting to share this journey with your daughter and grandchildren!!! Whatever you end up doing, I'm sure those kids will remember it for a lifetime!!

ORDTraveler, the part of the Oslo-Bergen train trip between Oslo and Myrdal is considered more scenic that the section farther west. By just staying on the train rather than dipping down to the Sognefjord, you skipped 3 of the 4 scenic components of the Norway in a Nutshell route; that's probably why it didn't meet your expectations.

I agree with you that skipping Norway on this trip is worth considering since so little time is available.

Thanks, acraven. We were aware of that at the time we made our decisions. We didn't have have time available for overnights and we knew that we would not be able to handle the full route comfortably in one day. I probably should have said above that after reading the descriptions of the Bergensbanen route, which is literally what we experienced, our experience was somewhat disappointing. Given that we knew we were skipping much of the full NIN, perhaps we should have just flown between Oslo and Bergen, but live and learn...

OP - There are also one day NIN loop options to and from Bergen to some of the more scenic areas without going all the way to Oslo.