Spinmetal is gotten from random plants around maps that glow silver. You can also get them from a certain exploration mission (I think assassination?).
You get extra reputation with a specific faction/vendor by using their gear when participating in the Crucible and I think doing bounties.
All this info is a bit hazy though since it's from the beta.
It would be super great if they explained that more explicitly. I've been avoiding doing patrols at all costs because I think they're incredibly boring, so maybe I would have picked that up if I was doing more of them.
Bounties and Patrols are basically the equivalent to Diablo 3's Adventure Mode. It's an easier way of levelling new characters and is how you prepare for the end game.
Someone needs to write to Bungie and tell them to take a closer look at Adventure Mode then.
Uh it is specifically called "Explore" Mode (unless they changed that for some reason) and all the missions you do give you vanguard rep unless you are wearing a faction class item then they give you rep with that faction. The only ones that give you spinmetal and big rep/exp are the assassination missions. All other mission types do not give resources and give less exp and faction rep.
Coincidentally Spinmetal is just the Old Russia material. The Moon material is called Helium Coils.
It's called "Patrol" in the final game. When you get down there Peter Dinklage gets out of bed at 4am to say "Now we can go anywhere." It would have been better if they explicitly said: "yo this is filler content" rather than having it look like any other mission on the map.