想係今年十二月去北歐自由行(3星期度)，Budget $20000內，有乜提議？

Hi there - Dec. in Scandinavia is quite cold as you would expect. Depending on where you are going you may get snow in some places. Denmark, especially Copenhagen, doesn't snow until January usually (although this year was unusually warm and only had 3 days of snow!) The more north you go, the bigger the chances of seeing snow. So bring a big, warm jacket - the wind chill factor is what makes the weather freezing.
You can do the big cities, starting either from Copenhagen or Helsinki, through Sweden and Norway. Let's see if we can come up with something here -
Finland (Helsinki) - 2 days, include a day trip to Tallinn (Estonia), just 1hr 45min ferry ride from Helsinki. You can stay overnight in Tallinn as well - there are quite a few things to see in this former state of the USSR.
Sweden (Stockholm) - 3 days. Can't tell you too much about Stockholm since I haven't been there yet but hearing from my friends who have been it is a very nice city, lots of canals and small side streets. It's like walking through a maze.
Norway (Oslo) - 2 or 3 days, then take a train to Bergen (7 hrs) - 'one of the most beautiful train lines in Europe' according to Lonely Planet.
Denmark (Copenhagen) - 2 or 3 days. The city can be done in 2 days if you don't go to museums. Most (if not all) the attractions are in the city centre and can be walked. Since it's Dce., there's not much sunlight (mostly raining lots in Copenhagen) and it gets dark at around 3pm.
There are a few things to see in Denmark as well that requires a whole day, for example 2 castles up about 1 hr north of Copenhagen would require a whole day since one of them is a National Museum. You can also go and see Hans Christian Andersen's birth house in Odense, about 1.5 hr train ride from Copenhagen (the ticket can be expensive though). Legoland in Billund is closed for winter.
If time and money permits, go to Iceland as well. You'd have to fly there and maybe join a tour group there to see the natural surroundings on Iceland. Many places in Iceland are hard to get to by yourself - you could drive but again, it's Dec. so it'll be snowing lots in Iceland, making driving very difficult.
Or alternatively, you can also take a trip up to Lapland in Finland to see Snata Claus. There are local flights provided by SAS or Finnair or Blue1 that are affortable.
Now one thing you should know about the Nordic countries is that they are quite expensive, especially public transportations. That is because the locals tended to ride their bicycles everywhere and rarely use the public transport system. The best way to see the cities is walking. In Denmark if you need to get out of Copenhagen, buy something called 'Klip Kort' (Clip card). This allows you to travel within Zealand, the island Copenhagen is located without having to spend almost double buying a ticket from the drivers. You can use that on buses, trains, ferries and metro within Zealand. To get to Odense, you'd need to buy a ticket from the ticket office at Central station.
Staying in hotels is expensive, hostels are a little cheaper. The last time I checked for a double room in Copenhagen for one night is around 725DKK (around 1024 HK dollars) without breakfast. Single rooms are a bit less expensive. Check Danhostel for more info as it's a highly recommanded place.
I would suspect Oslo to be more expensive than Copenhagen (it is the most expensive city in the world after all!), though Sweden is about 20 to 30% cheaper than Copenhagen. Finland is about the same as Copenhagen. Tallinn is very cheap compared to all the other places.
So your budget might be a bit tight, meaning that you may not be able to go to so many places due to the high cost. Don't forget eating out is expensive, too, without not many choices either!
Good luck and wishing you a safe and fun trip.