I believe the airbrush machine that I just got for my birthday is the same model as yours. (I think it was purchased from Amazon, too) I understand your frustration about how to assemble the machine. I have never used an airbrush machine either and you're right...there aren't any instructions! I can happily say that I have used this machine several times so I think I assembled it right.
Basically, I connected the blue braided cord to the airbrush pen and the other end to the black valve type thing. You need to screw both ends on. You connect it to the "side piece" of the black valve type thing. (Sorry, I don't know the technical names for this machine.) You will squirt the paint color into the round canister attached to the airbrush pen. I was able to find some YouTube tutorials that were helpful.
PM me if you need a better explanation. I can probably post a picture of it assembled so it's easier to understand. Good luck!