As much as I fault Destiny for its extremely repetitious and grind-heavy end-game, there is no denying that the game truly does start once you hit the soft level cap of 20. Sure you stop earning XP toward your overall level, but it's at that point you start earning some truly powerful gear imbued with Light, which raises your level past the game's maximum.
But there are actually completely new activities available to you once you hit level 18 and higher, some which the game doesn't do a great job at explaining. So with that foreword, here are things you should be focusing on when you're level 20.
Join a faction, or don't
Did you know there are three factions to join in the Tower? Not everyone does. In fact, many people run by these vendors and just think they're a level 20 vendor, without realizing those items never become available once you hit the level cap.
Each faction has two stat specializations for armor. New Monarchy focuses on Intellect and Strength, Dead Orbit focuses on Discipline and Strength, and Future War Cult focuses on Intellect and Discipline. What this mean is, you are basically picking out which of the two primary three skills you want cooldowns to be shorter, your Super Charge (Intellect), Grenade (Discipline) or Melee skills (Strength).
Once you pick the right faction for you, you'll need to spend some Glimmer on one of the two Class Gear items. Once you wear them, any activity which would normally earn you Vanguard reputation, will now earn you faction reputation. However, all the gear in each faction also requires Crucible Marks as currency, so you'll have to mix a good dose of PvE with PvP.
If it's just PvE or PvP that you want, without having to balance the two, you can stay a Vanguard, and earn reputation and marks through PvE events such as Strikes and Patrols, but I'll get into specifics in a bit.
Earn your Mark
One of the most important end-game currencies are Vanguard Marks. These can only be obtained through the Strike Playlists, Daily and Weekly challenges.
At level 20, you can select the appropriate Strike Playlist based on level, so a level 20 playlist will earn you 2 marks per Strike, 22 will get you 3 marks per Strike and 24 will get you 6 marks per Strike. If you're with a competent and leveled group, you can breeze through these no problem. However, keep in mind that there is a cap on Marks for the week. You can only earn 100 Marks a week, and considering some of the gear requires 150+, you'll need to at least play for two to buy that one item.
Daily and Weekly missions also reward Marks upon completion, so completing these is also important if you want to boost your Marks.
Level beyond 20
Destiny has a unique way of leveling your character once you reach the level cap of 20. You no longer earn XP and level up the traditional way. Instead, you earn gear that has a Light attribute attached to it. This Light score will add up and once a certain number is reached, it will ascend you to the next level.
There are a few things to keep in mind. Only Blue (Rare) armor or better has an associated Light attribute. That means, anything Green (Uncommon) you get is essentially garbage and you should dismantle it into materials immediately, since materials are important when upgrading your gear.
What's also important to note is that the Light attribute actually goes up when you upgrade that item's defense score, so you have a chance of gaining a level without actually changing up your equipment, and just making sure it's fully upgraded.
Go on Engram/Glimmer Runs (while they're still available)
While Destiny is essentially a loot game, it doesn't really serve it up in troves. However, there are a few methods to earn Engrams and Glimmer in a few locations.
Now, keep in mind, since Destiny is a constantly evolving game, much like an MMO, there is a good chance that Bungie might patch these methods.
Engram Runs consist of going to locations with a heavy monster population that either respawns really quick, or allows you to kill yourself and be respawned with all the monsters coming back as well. As far as Engram Runs go, there are two listed here, both on Venus, though one you'll do on Explore mode, while the other you do in Venus' Strike, Nexus.
Glimmer Runs are actually important, if not more than Engram Runs, since they allow you to level up the Cryptarch quickly and thus earn Legendary items. There is one Glimmer Run as of right now, which I posted about earlier right here, and takes place on the Moon, during the Shrine of Oryx mission.
Keep your daily checklist handy
Once you hit 20, chances are you'll be repeating a lot of the same stuff over and over again. The game does give you some incentive to doing so, by way of Bounties, Daily and Weekly challenges.
Bounties are pretty important, as they're your primary way of earning reputation. Much like Diablo 3's Adventure Mode, they'll require you to revisit an earlier location, and complete various challenges like kill 30 enemies with your melee strike without getting dying, kill a certain boss on a harder difficulty, kill two enemies with one shot of your Fusion Rifle 20 times, etc. The bounties refresh each day, so after you're done with your current ones, you'll have to wait to take on more. However, there are also Crucible bounties, which increase your Crucible reputation, so you can buy some PvP gear from the Crucible vendors.
There are also Daily missions, which are harder, level 20+ missions that you've already beaten. They usually consist of rewards like XP, as well as gear and some items. Then there are Weekly challenges in the form of Weekly Strikes and Nightfall Strikes. The Weekly Strikes are extremely hard versions of regular Strikes with increased damage and health pools for every enemy. Nightfall Strikes should only be attempted by level 26 characters that are extremely well coordinated, as enemy damage is through the roof, as well as their health pool, and if all three members die on the mission, it's back to orbit!
Strange Coins? Motes of Light? Wha–?
When you reach level 20, you technically still earn XP, as that's the only way you upgrade your arsenal and gear, as well as unlock new skills on your skill tree. And while you no longer see your XP bar at the bottom of the screen, you still do earn XP for your level, except now instead of gaining an actual level, you'll earn a Mote of Light. This currency can be used at The Speaker on the West side of the Tower. His items however only consist of Class Gear (like the cloaks for Hunters) that have no attributes and Player Emblems.
As you complete various daily activities and find chests around each planet, you will come across Strange Coins. This currency is used for the Weekend merchant, Xûr, the Agent of the Nine. His stock consists of Exotic weapons and gear, and requires Strange Coins in exchange. Xûr will appear starting Fridays until Sundays, and his stock remains unchanged for the entire duration of the weekend.