If you're anything like me you probably already have a bag full of weapons of various types, if not you should be able to buy whatever you're missing, then simply get a kill with each:
- Assault Rifle (ASR)
- Submachine Gun (SMG)
- Shotgun (STG)
- Light Machine Gun (LMG)
- Sniper (SNR and DMR) (two separate types of guns)
- Pistol (switch by double tapping
)
I didn't start seeing shotguns until a little later, but that's probably just random.