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 Rifles:
Braton - Market for 25k credits
MK1-Braton - can be taken in the tutorial mission, if the player chooses a bow this rifle can be bought in the market for 15k credits
Common Rifle Blueprints:
Burston- Market for 15k credits
You can track the current active alerts using
https://deathsnacks.com/wfxb1/