Some useful info on farming Reputation for Vanguard/Crucible/Factions:

Do the Nightfall BEFORE the weekly heroic. When you complete the Nightfall strike, you get a buff which increases all experience gains and reputation gains (This includes cryptarch) by 25%.

The nightfall itself gives 500 rep to whatever faction class item you have equipped or Vanguard by default if you dont have anything equipped.

The weekly heroic strike gives 400 rep by default, which then goes up to 499 rep (I dont know why its not 500) if you have the buff.

Rep is also gained by decrypting engrams at the Crytarch.  You should save engrams to trade in so you gain more Rep:

Color / Unbuffed Rep / Buffed Rep
Blue Rare / 150 rep / 187 rep
Purple Legendary / 200 rep / 250 rep
Exotic Yellow (from Xur) / 500 / 625 rep

You also get it for buying white/green items from the Crytarch:

Color / Unbuffed Rep / Buffed Rep
Green Uncommon / 10 rep / 12 rep
White Common / 5 rep / 6 rep

Only buy these at high level when your maybe less than 100 rep from a level, or else you waste too much money. It's most glimmer-efficient to do this is with the 450 glimmer green engrams. It nets you the most rep/glimmer plus you also get weapon parts/upgrade mats when you dismantle.

In case you don't know, after rank 3 cryptarch, you get a package in the mail which gives either a few blue/sometimes legendary engrams, some glimmer, and some rank for cryptarch.

You need 3000 rep every level after this. Unbuffed, this is 20 Blue engrams. Buffed, it goes down to 16. This doesnt account for the rep you get from the package itself or any legendaries/exotics you may find and hand in. It really does help doing the nightfall before ANYTHING after weekly resets.

Reputation buff does not apply to Queen's Reputation. Queen's rep will go up 10% once your Rank 1, and then another 10% I think once Rank 2 for a total of 20% additional gain at Rank 2

5 types of Patrols:

1. VIP - Star emblem - 25 rep
2. Kill All - Exploding Ghost - 10 rep
3. Gathering/Resources - Triangle/Pyramid - 10 rep
4. Scan - Ghost emblem - 10 rep
5. Scout - Compass - 10 rep

This farming route is on Earth: Cosmodrome

How it works:
1. 3 Man Fireteam, Split Cosmodrome into 3 parts, each member takes a location - The Skywatch, the Mothyards/Steppes, The Divide/Rocketyard - Stay spread out to be able to cover as much area per player as possible, overlapping potential objective areas is potential time lost

2. VIP's spawn in Rocketyard, Skywatch, and the Steppes by spawn - never abandon VIP or Kill Everything patrols. Others can be abandoned if nobody is near the objective. The spawn of the patrol isn't always near the actual VIP, so communicate who actually gets the kill.

3. Farm patrols vigorously. Triangulate position of objectives and have one member complete objective as the others farm their respective areas (Skywatch farms Loot Cave, Rocketyard farms chests, Mothyard farms Spinmetal). Abandoning patrols is okay, but try to minimize as it is time lost farming - only abandon stubborn ones or ones that nobody is close to quickly do and they can be replaced.

4. Allow players to do nearby Public Events if you need marks. Pay attention to the number of other players doing the events, because soloing them slowly can cut your numbers in a big way.

5. Learn to communicate very well to improve rep/hour numbers, because good communication allows you to locate objectives quickly and to know when to abandon a patrol for a more accessible one that is readily available. Most notably, communicate when to go for objectives and when to be farming per member. Pay attention to what types are near you even when you're not going to be picking it up first, so that you can know to replace a stubborn patrol quickly with the easier one you have nearby.

6. If you really want to solo, farm VIP spawns by the spawn point into the Cosmodrome at the Steppes. It's to the left of the basement with the 3 level 18 Hive Majors (used to kill everyone in beta).