Upon reaching Level 18 you unlock the ability to gain vanguard and crucible marks.
The maximum weekly score is 100 (same for crucible marks).
Vanguard marks are unlocked in the following 'missions':
Daily Heroic Story, Level 20 Story, Rewards: +5 Vanguard Marks
Strike Playlist;
Level 20 Strike, Rewards +2 Vanguard Marks
Level 22 Strike, Rewards +3 Vanguard Marks
Level 24 Strike, Rewards +6 Vanguard Marks
Weekly Heroic Strike;
Level 22 Strike, Rewards +3 Vanguard Marks
Level 26 Strike, Rewards +6 Vanguard Marks
Level 28 Strike, Rewards +10 Vanguard Marks
Public Events, +4 Vanguard Marks per completion and +2 Vanguard Marks for first daily completion from postmaster.
Converting 50 materials, +5 Vanguard Marks (Foxfire49)
I really recommend not converting your materials as you need them to upgrade legendary and exotic gear so for only 5 marks it really isn't worth it.
Weekly Reset occurs Tuesdays.
Personally I find the fastest way is to play the level 24 strike playlist as I get through each one in 15 - 20 minutes, pairing that with the weekly strike, daily story and public events will achieve this in no time.