Do I need to buy the real Skyrim game and the Dragborn DLC for the Dragonborn DLC to work? Or just the real Skyrim game?

You need both. Bethesda may offer a version of the game later on that includes the DLCS, but for now you have to buy the base game and the DLC that you want.

I would suggest you get the legendary edition that will be comeing out soon. It's more for your money. http://www.bethblog.com/2013/04/25/legendary-release/

Actually the Dragonborn DLC partially works for me, but the only problem is that I can't travel to the island? I didn't buy the Skyrim game and the Dragornborn dlc, I downloaded them in the internet instead because it was free. Could that be the cause of the problem?

Um...from the sounds of it, you are literally admitting that you pirated Skyrim on the company's own official forums. That's generally a bad idea.

Closed due to obvious stupidity.