Copenhagen and Bergen are separated by a distance of about 1,000 kilometers (620 miles), which can be covered in several different ways by travelers. Each transportation option between Copenhagen and Bergen has its own pros and cons.
Copenhagen to Bergen by Air
Taking one of the hour-and-a-half direct flights by SAS or Norwegian is definitely a good option if you want to get to Bergen or back to Copenhagen quickly. The prices for a one-way ticket aren't too expensive if you book a bit in advance. The ticket prices will add up if you need round trip tickets for the entire family, though, so other transportation options might be more suitable.
Copenhagen to Bergen by Train
It'll be a long journey if you want to take the train. There is no direct rail connection between Copenhagen and Bergen. You have to travel via Oslo and you'll need about one and a half days (still not as long as the bus option, though). You can book tickets on RailEurope.com. Getting from Bergen to Oslo by train takes seven hours alone but it is a scenic journey and the best option for travelers who have time to spare. A Eurail Pass ticket for Scandinavia is another more flexible, cost-saving option.
It is reasonably priced and can cover Denmark, Norway, and Sweden all at once.
Copenhagen to Bergen by Car
Quite a few travelers just rent a car. There are two main routes taken by drivers between these cities, and the travel time of 12 to 13 hours is the same for each route. The more scenic drive takes you across the Øresund Bridge and through Malmö. Then, follow the E6/E20 north merge onto the E16 to Bergen.
The second option involves the short Helsingør/Helsingborg ferry (which departs frequently) and the same road heading north thereafter. To drive from Bergen to Copenhagen instead, simply reverse the directions.
Copenhagen to Bergen by Bus
If you're focusing on keeping your travel budget as low as possible, nothing beats the bus. Yes, it's slow. Yes, it can be uncomfortable. But it's also the cheapest transportation option and can be relaxing. Keep in mind you'll spend 18 hours on the bus to get from Copenhagen to Bergen and that doesn't include waiting for your connection.
Swebus Express bus line 820 connects Copenhagen (select bus station named "Köpenhamn") and Oslo. Between Oslo and Bergen, use Nor-Way Bussekspress, which offers daily morning departures in each direction.
Copenhagen to Bergen by Ship or Ferry
Pricey, but pretty. This is a beautiful ship journey, assuming you want the Copenhagen-Bergen connection to be the focal point of your visit. Scantours and CruiseDirect even offer complete cruise tours that have both cities as ports of call and can be joined for the Copenhagen-Bergen trip. Pick this option if you have a decent travel budget and several days available.
On a cheaper note, you can also just book the Copenhagen-Oslo ferry and then travel from Oslo to Bergen with a different type of transportation.