You do not get experience in the vault, crota, or the prison. You only get exp in crucible for 2 reasons: 1) kills with said weapon 2) crucible mode that has exp bonus(usually skirmish) 3 lvl 34 characters... Doesn't know end game content doesn't give exp at this point...

Hey mate, the fastest way is to do all bounties on all 3 characters. Get a rocket telemetry (gunsmith, but it's not necessary) activate it right before handing in the 10 bounties and make sure the weapon is equipped and nothing else that can be levelled up including armour or you will lose some xp to that item as well, then do the weekly or daily mission within the time limit. Do this for all 3 characters and it will take around 30 bounties to max. If you don't have 3 characters you will have to wait until daily reset and go again. Hope this helps.

Raids don't give xp

Go to the gunsmith and buy 2-3 rocket launcher telemetry packs, 250 glimmer & 5 weapon parts, he doesn't always sell so get while you can. Collect all the bounties you can, do them all, equip a rocket telemetry before you hand them in, this lasts 30 minutes. Be sure you've got the gjallahorn equipped. Hand in the bounties and upgrade as needed. Go & do daily story with time remaining.