The quickest way to level a weapon is to play Survival or Defense missions. Only keeping the weapon equipped you are leveling can make it go quicker.
Warframe power kill = 100% affinity to warframe
Weapon kill = 50% affinity to warframe, 50% to used weapons
Shared affinity (team kills, etc) = 25% affinity goes to your warframe and 75% divided to loadout. eg If you have just 1 weapon it will get 75%, if 3 weapons 25% each
Common Shotguns:
MK1-Strun - Market for 15k credits
Common Shotgun Blueprints:
Boar - pretty common drop from Drekar Troopers (Mastery Rank 2)
Hek - Market for 40k credits (Master Rank 4)
You can track the current active alerts using
https://deathsnacks.com/wfxb1/