I capped my Assault Rifle Skill to 20 by doing a lot of co-op missions. If your main focus is to skill up, avoid Arkfalls because Hellbugs and Scrappers don't seem to give any weapon xp and enemies that do give weapon xp generally do not last long enough for you to continually do damage on.
Co-ops, main missions, side missions, and emergencies are the way to go for skill ups. Remember to keep plenty of LMG's on you, you'll want to swap them as soon as weapon xp caps.

