Each objective within a Mission set has an XP reward, and sometimes another cosmetic or weapon blueprint reward.

Do you get weapon XP for objectives?

Combined Arms games also centre around objectives, which will make it easier to stack killstreaks and earn even more Weapon XP if you can find a good objective camping spot, or take advantage of a major route that enemies use to reach the objective.

What gives you weapon XP in Modern Warfare?

Get experience points (XP) every time you kill an enemy. XP not only goes to your rank but also to your currently used weapon. Your weapon levels up once you reach a certain amount of points, which unlocks new attachments and camos for you to use.