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Many of you (if you haven't done any research) will be a bit confused by the artifact leveling process. Yes you can level up your alt spec artifact weapons, but for RAIDING you'll really want to focus on one weapon.You'll be getting items from nearly everything with loot that give artifact power. How much you get depends on the research youve done from your garrison..erm Class Hall. The more research missions youve completed = the artifact power gained from items.

I STRONGLY suggest watching this video:
https://www.youtube.com/watch?v=_6u4OKWgAgs

Make your decisions wisely...

FOR VOL'JIN!!!

Basic gist is that your artifacts require ever more ridiculous amounts of power to add talents, which is why you can keep your offspecs at about 80% of the main weapon (though you may not want to do this if you're really interested in pushing content).
The other big takeaway is that in order to maximize your artifact power you should be doing research missions as often as possible to get the bonus multiplier to power pickups.

Based on what we have from beta and PTR, this is a representation of where you'll stand in artifact power if you focus exclusively on 1, or work on 2, after 28 days, assuming you do all dailies and possible research missions post-haste: http://imgur.com/ZUUThTR
The numbers change if you try to do 3 or 4 simultaneously (probably not recommended).