These drop more easier than the original Destiny, by completing a Crucible or Strike activity you will receive one or two drops respectively. Playing the fast-to-complete competitive Crucible matches will enable you to unlock Weapons and Armour quicker but completing a Strike will give you two loot drops compared to the Crucible's one and will also include some exclusive armour.
You can only get Weapons from completing activities, there are no weapon drops from Engrams or from vendors. There is no need to keep duplicates as all individual weapon and Armour types have the exact same stats and light level. Each of the characters comes with a set of Kinetic, Energy and Power weapons that are already unlocked and further variants of them will drop as you complete activities.