I started playing with the free giveaway they did around black friday I think it was, and here is what I've done to get to ~PL640 on 2 characters without any pre-made fireteam and no clan.
From 500-540ish you will gear up fastest doing heroic public events. Just make sure you keep (ie, don't dismantle) your highest PL item for each slot as the game will consider this when looking what PL item to drop for you. Look for the weekly flashpoint and follow the active public quests around, for the first couple of days after a reset these will be quite busy. Just read up a little on what makes an event heroic so you don't screw the event accidentally.
After about 540 you are going to be looking at completing the daily and weekly milestones that award 'Powerful Gear' as these are the main source . These are marked in the Director with a couple of icons (daily = a 4 pointed star, weekly = like a bright gold star). Use
http://www.destinyreset.net/ to keep track of when stuff is going to reset, although the weekly reset is always on a Tuesday. Also you will automatically get a buff called 'Prime Attunement' which will up the drop rate of prime engrams until a certain number of them have dropped, which will also drop items above your power level.
Also from 540 you can complete some of the adventures and missions in the Dreaming City but it is noticably harder than other areas of the game imo.
Note that Iron Banner when it is on has weekly bounties that also reward Powerful Gear. This is a good way to play catchup as the quests aren't hard, they just require you to play a fair amount of rounds (how many will depend how good you are at PvP)
At this point one of two things is going to happen. Either you will run out of time each week to complete them, in which case you are going to be capped in how fast you can progress by your available game time, or you will run out of 'powerful gear' type milestones before the reset. If the second thing happens you can create more characters (as they each have their own milestone lockouts) or look to do more activities in a group such as Nightfall Strikes and raids.
On top of this there are many quests that will reward you with exotic or pre-canned legendary items. These range from relatively straightforward to almost impossible for normal players. Something like Cayde's Will for the Ace of Spades is a decent quest to start with, don't be put off by the PvP steps if this isn't normally your thing. I suck at PvP but managed this and The Last Word quite easily. I've just finished Sleeper Simulant which whilst not hard, was quite grindy!