When you talk about Conan and Elric, you are really talking about three different characters. This is because the Conan in all of the non-howard stories is very different than the Conan of the howard stories. In the howard stories Conan is a commentary on society. He shows that the civilised can oftentimes be more barbaric than the barbarian. He shows that civilisation is not always clean, sane, or safe. And Conan is a feminist (at least by 1930's standards). The Conan that came after Howard's death is a completely different person.
The interesting thing about Conan and Elric is that while they come from the opposite ends of the spectrum and one is the opposite of the other, they both do the same thing. Elric is just further proof of the thesis that Conan is built on. Elric shows that the civilised are capable of extreme acts of brutality, insanity, and dangerous decadance.
As for the storeies to read about Fafhrd and the Grey Mouser:
Ill Met in Lankhmar is a good starting place
Thieves house
Claws in the night
The Price of Pain Ease
Bazzar of the Bizarre
The Cloud of Hate
Lean times in Lankhmar
The Witches Tent
The Two Best Thieves in Lankhmar
The Lords of Quarmall
The Swords of Lankhmar
I have read up through Swords of Lankhmar and these are the ones that I find define the characters the best.
As for which one I voted on? Conan. As a character his cunning knows no end. He has had lots of experience with the Elric type. While he is violent, he does plan and contemplate. On the literary side, Elric is a reaction to Conan. And Fafhrd is a shadow of him. So in the end, its Conan.
Aaron.