No, the US is not trying to provoke a civil war, that would not be in their best interests. Though its essentially whats going on now, the insurgency in Iraq doesn't focus much on the invaders, it attacks other sects and the Iraqi state. Increased instability is what provides them the opportunity to form their own secterian government.
Why do you want military experience? All that means, really, is that you killed brown people for state and capitalism. That isn't something you should want.
I first, would advise against joining the military. Especially the legion as their stated purpose is to enforce french policy in the colonies and it is comprised of some terribly racist elements.
I was in Iraq, the resistance is definitely majority Iraqi, not foreign, and certain groups enjoy various ammounts of support, but no group is universally supported by the whole country. The shias and sunnis each have their own groups that they support, and sometimes groups from the same sect will fight each other as well.
The resistance in Iraq is as secterian as it is anti-us, and there are no clear groups that I know of to really support. (thats not to say I support imperialist victory though)