The fastest ways to rank up vanguard are: 1) play heroic strike playlist with the rep booster active. Add on a vanguard booster class item and I think the rep is something like 240 or 250 per strike. The strikes usually only take 10-15 minutes, so you can get around 8-12 strikes in during the 2 hours for the booster, which is enough for a rank up. There are a lot of posts on here complaining about how crappy the random teammates can be in the strike playlists with people going AFK or not actually playing the strike. Either those posts are the exception or I've been very lucky. The people I get paired up with the in strike playlist are usually active and pretty good. 2) go crazy with the bounties and patrol missions. Looking at today's bounties, there are 3 that give 65 rep and 3 that give 50 rep. That 345 rep there, plus another 10% if you wear the vanguard boosting class item. Then while you're on the planet, pick up and complete all of the easier patrols. those give 40-50 each.

I know this helps you out diddly squat but I just recently got the Badger and Hand of Judgement - both with exp rounds and some other nifty perks. Both guns are pretty similar but damnnnnnnn......they are SO good to use. I was always an auto fan...but my go to guns are now those two. Completely changed the way I play the game in PvE and PvP. I hope you get your hands on one...I really do. God I think everyone should get a decent Badger CCL....possibly the most underrated legendary in the game IMO.

Grab every bounty from the Tower. You can very likely get half way to Vanguard doing the Tower Bounties and doing the bounties in the areas you are patrolling to complete them. For extra bounties (patrol bounties), make The Moon a patrol fixture. The Moon will not have Taken Events to interrupt your bounties, so you can get more done within a certain time. Unfortunately, Vanguard Bounties do not seem to have a quick fix like the faction bounties (motes and materials).