Arguably the most significant change in Destiny with the arrival of Rise of Iron is that every End Game activity is now accessible at the highest level for all players, regardless of taste. Right up to and including The Taken King, only specialist activities with "Don't you dare suck!" messaging hidden between the lines, could get you to the level cap. Meaning that, unless you were prepared to study and practice and yet still fail for hours on end during Hard raids and the near-impossible (for mere mortals) Trials of Osiris, your light level would prevent you from joining with friends whose patience was greater than yours. Probably unfair...
Now, all you need to reach Light 385, is time and a bit of a clue. And some more time.
To provide a very short and simple answer, the only legitimate way to reach Light 385 is to play the strike lists and crucible - in particular Iron Banner and Trials of Osiris - and wait for decent loot drops. Combine these activities with Faction rewards, achieved by earning EXP on relevant Bounties, and better loot will be yours as a matter of course. As a general rule, always equip your highest Light level gear to influence the item drops during PvE activities, and boost the likely Light level of items decrypted from collected Engrams. Nothing much changed since The Taken King in most respects. However here are some points to consider:
Assuming you're starting from Light 335 or close enough, your first chances to make a big increase arrive after completing the main story missions. Tyra Karn and Shiro-4 have Light 350 gear, in the form of Class Item and Iron Lord Artifact respectively. The latter requires a simple side-quest to complete, but it's soon done. Shiro-4 just needs a Legendary engram, plus 2500 Glimmer. Alternatively, you have Legendary Marks to turn to, but spend wisely.
Legendary Marks can be exchanged for armour and ghosts and weapons at Light 350 from Faction representatives in The Tower. You'll use these up fast, so it's better to buy a ghost since these are rare drops during activities. The Speaker sells Class Items at 350 also.
While you're climbing toward Light 340 the most valuable use of your time is the SIVA Crisis strike playlist at Light 320. Make sure you don't return to Orbit (exit) unless necessary, to benefit from the bonus streak that increases the chances of legendary drops. Normal SIVA strike loot drops up to Light 340, rare engrams also decrypt up to 340. Plenty of these drop during the strikes and after the bosses have been defeated. Legendary engrams eventually decrypt up to 365, but only ever cash in between 1-5 levels above your current overall build. Xur is also another viable shortcut, offering Light 350 exotics that you can purchase to equip or infuse into your existing legendary or exotic gear.
As soon as you're able, hop into the SIVA Heroic strike playlist, where rare items drop up to light level 365. Unfortunately, and infuriatingly, rare engrams are capped at Light 340. Keep on equipping the higher level armour and weapons as soon as they drop to ensure that all subsequent drops have a chance of being higher still. This is the route to 365, pretty much.
After a while you'll notice that things are slowing down a bit, and the reason is most likely that your artifact is holding things back. Rare and legendary engrams of all kinds have a chance of decrypting into artifacts, fairly sure one of ours came from a helmet. Otherwise, Eris Morn occasionally rewards artifacts for levelling up. If you fancy a break, head to the Dreadnaught and the Court of Oryx to unlock an Eris Bounty and pursue this to the end. Completing the Challenge of Elders in the Reef gives you a shot at 365 items from Variks. Your first gold tier Public Event of the day rewards a guaranteed legendary engram, but remember the max these reach upon decryption is Light 365.
If you've stuck with it this far, probably 20-25 hours, you won't mind how things go from here on in. Basically it's SIVA Heroic strikes, plus as many Three of Coins as you can afford. Strike specific items such as the new Devil's Dawn sniper rifle and Hood of Malok drop at up to the existing level cap of 385. Exotic engrams decrypt up to 385 also. We'd like to suggest heading to Archon's Forge in the Plaguelands for chances of (up to) Light 385 item drops, but it's not always guaranteed that you'll find fellow guardians hanging around there. If you've a fireteam with you, and a SIVA Offering or three between you, go for it.
Last and not least, there's always the Wrath of the Machine raid. But nobody's forcing you. It's probably no big deal anyway. Just say you've done it. Or just do it for something to do.