1- Copenhagen to Bergen by Air
Taking one of the 1.5 hour 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 one-way ticket start around EUR 70 (US$100), if you book a bit in advance. (The ticket prices will add up if you need roundtrip tickets for the entire family, though, so other transportation options might be more suitable.)
2- 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 1 1/2 days (still not as long as the bus option, though). Getting from Bergen to Oslo by train takes seven hours alone but it is a very, very scenic journey. The best option for travelers that have time to spare. A EUrail train ticket is reasonably priced and can cover Denmark, Norway and Sweden all at once.
3- Copenhagen to Bergen by Car
Quite a few travelers just rent a car locally. There are two main routes taken by drivers between these cities, and the travel time of 12-13 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 every 15 minutes) and the same road heading north thereafter. To drive from Bergen to Copenhagen instead, simply reverse the directions.
4- 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 very relaxing. Keep in mind you'll spend 18 hours on the bus in order 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.
5- 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.