Which is better for progressing as quickly as possible in diablo 3, killing everything you can with a +xp gem in a helmet, or completing missions as fast as possible, while just killing monsters that you encounter along the way?
It depends :)
If it's your first character, you should probably take the 'Kill ALL the things!' approach, or you could find yourself starting Hell/I with lousy gear and no money to upgrade it with, and have to go back and kill all the things anyway.
For subsequent characters, I definitely prefer to rip through the quests as fast as possible. Success in Normal and Nightmare in particular are far more dependent on gear, and you'll get enough XP from just doing the quests and only killing the stuff that's in your way to keep from getting too far ahead of the curve. I've done this three times, and the only real speedbump I've hit is only being in the late 40's at the end of Nightmare/IV, and having to grind out XP until level 50.