With The Taken King out for a week, it's good we try and see what's new and better in the Destiny universe, helping you in the new things Bungie has shaped up to its shared-world shooter game. Among the most different things you will find in the title, in comparison with the original experience, surely there are light levels and questing.
Previously, Blue items were capped at certain (low) light levels. This changes with The Taken King, as you can now get Blue drops up to 300 light level, making them often better than Purples you'll find. The higher light level you currently are, the higher the light level of the blue engram you decrypt. For this reason, it's best for you to decrypt engrams one at a time: decrypt it, go to your inventory, check it, if it's an upgrade then equip it, go back to the Cryptarch and decrypt the next one. As you continue to do this over the course of 50-100 engrams, your light level will edge up and up towards 300.
The Taken King is very different to House of Wolves as it requires significantly more time to get your character to a point where they can only be advanced through weekly reset activities. For that reason, it's best to put all of your effort into a single character from the get go. Doing so will significantly increase the speed at which you start getting 300 light level items. These items can then be passed over to your alts to significantly increase the speed at which they progress.
Unquestionably, the fastest way for you to do max out your light level is by running level 36 strikes. Your main aim to gain light level is to maximize the number of engrams you can decrypt, and the fastest way to get these is in level 36 strikes. At this point, it does not appear that Heroic strikes reward additional rewards over and above what you would get from the Lv 36 playlist, so it does not appear to be worth the extra effort (note: this is still unconfirmed). To max your light, just run Lv 36 strikes over and over.
In short, if you are chasing light level as soon as possible, spam strikes and don't quest. If you are wanting to access exotic quest chains and don't mind a slower path to higher light levels, do quests. You'll spend a lot of time waiting around etc while doing this, but completing the main story and secondary quests you will get engrams along the way. Quests do not provide you with decent loot (with rare exceptions) so don't rely on quest rewards to gear up.
They do however set you up for end-game quest lines for exotic items, so while this route may be slower in terms of getting your light level up, it's faster as you work towards completing the meta-game.
You'll get a quest that asks you to complete 5 lv 36 strikes, and then another to complete 5 heroic strikes. After that, it asks you to do a Nightfall. It's worth doing this because it rewards a 300 Light Shotgun. Don't do the heroic strikes until you are 270 light. Doing it at 260 will match you with other 260 players, and the strike is extremely difficult at this point.
You are now able to increase the light level of your Purple items through a process called Infusion. In short you take a stronger item than the one you want to infuse, and you then break it down to make your weaker item stronger. This is where a lot of your endgame currency (Legendary marks) will be going.
Ascend your Sword to 280 as soon as you can: once you have done this and leveled the Sword up to its max perks, you'll unlock the next step in the question chain, putting you on a path toward an Exotic Sword.
When you buy an exotic from Xur or the Kiosks, they will be 280 Light. The max light level in the game currently is 310. Your aim is to get these items to their max light level as soon as possible. Purchase these items and use your blue items to ascend them.
Credit goes to reddit user Vespas