There are a couple of training missions you will have to do.
First of all it is the rifle range. you'll need 24 of 40 to pass (36+ and you unlock sniperschool). My tip is that you never shoot more than 1 shot per target, as you might miss the second round too, and thus basically missing two targets. Never fire the rifle until the sights are still, when you hold your breath.
Second mission should be the obstacle course. Listen to the instructions and you should pass it in a matter of minutes.
The third mission is the weapons familarization. Pick up the guns and fire them. Exit the course and you are then allowed to play several online missions.
There is a fourth mission, a CQB-training mission, I don't think it is mandatory, but it is farily simple as well.
POZBURN: do you really need 20 honor for SPR? 15 is the lower limit for Special Forces.