I like to mix up weapon loadouts anyway, so the weapon challenges were overall the least painful for me to get out of any of the categories, more or less.
There isn't much of a secret, although people have posted their favorite ways to try and make them go faster. Try to match the weapon with a character that uses it well. This does not necessarily run GI one hundred times for all the weapons, but I would recommend using more weapon damage friendly characters for weak guns, and then whoever for stronger guns. Also keep in mind characters with racial (multiplicative) weapon damage bonuses.
Really the only oddball mechanic that is helpful is that Sabotage's backfire xp is awarded to whatever dealt damage to the target most recently before backfire hits. This is why backfire kills sometimes go to different players, but it also means that you can get a little extra gun XP if you run a character with Sabotage (QFI makes the most sense) against organics. This is nice for some of the weak guns to get their points a little faster. I would use that mainly for weak guns that are relatively accurate, since inaccurate guns work better on accuracy boosting classes like Destroyer or Turian Soldier.