No need to fly, but if you are short on time it might be helpful.
From Cologne you can take a train to Copenhagen, it takes about 9 hours during the day, with just one change.
I guess there will be buses as well, likely overnight ones, as that is what I remember from older timetables. But the overnight (no change) train is no longer on the schedule.
From there you can easily travel to Sweden and from there to Norway.
I would do it in that order, as I did it the other way around and was disappointed by the tame nature of Sweden. But if you prefer culture over nature, keep Sweden or Copenhagen for last.
If the ferries between Norway and Denmark run at the time you want to travel, you can go back south that way and see mainland Denmark on the way.
Do not just to cities, in Norway you should go to west coast, the fjords. If you have enough time you might like to go north as well, see the midnight sun or northern lights depending on the time of the year.
I would not do such a travel with less than a week per country, more time will be better.
If you have only one week and that one in winter, select one location and travel there as efficiently as you can, like flying to the north of Norway to see the Northern lights. Or only Stockholm or Copenhagen.