In all of Bethesda's ES games, finishing the main quest DOES NOT end the game. So you can finish the MQ in Skyrim and still play to your heart's content. I remember when I used to play Morrowind, I didn't even know there was a Main Quest until about my 3rd character and many, many hours of game time.
BEATtudorsee, the reason nothing happened when you killed the dragon is probably because you have not read any word-walls with the dragon language yet. The souls only do something when you "learn" a new word or phrase, and some words require more than one soul to activate. The "dragonborn" is supposed to be an innate ability and is not something you get because you did the main quest.
I would suggest doing the main quest up until you kill a Dragon outside Whiterun (basically the 2nd MQ mission) because it is a good stopping point, you get a dragon shout, and a companion (if you want one). Thats what I did, and I haven't touched the MQ since.