Destiny 2

Each class in Destiny 2 has three subclasses, unlocked as you progress. For clarity I will split them into three groups:

1) Destiny 2 subclasses;

Hunter - Arcstrider
Warlock - Dawnblade
Titan - Sentinel

These subclasses are brand new for Destiny 2, and are unlocked very early in the game in an unskippable story event.

2) Vanilla Destiny subclasses;

Hunter - Gunslinger
Warlock - Voidwalker
Titan - Striker

These subclasses are from year 1 of the original destiny, but have been modified. To unlock these subclasses in Destiny 2 you must first reach level 7. At this point you will have a chance to find artifacts in chests in patrol. These can be gold regional chests, or the reward chest for completing a public event or HVT encounter.
Upon acquiring this artifact, you will revieve a new milestone instructing you to charge your power by defeating enemies with fellow guardians in public events, however I noticed the percentage went up from kills in any activity, even when solo.
After reaching 100% on the milestone, return to where you obtained your first subclass for a short, simple mission which I will not spoil here. You now have a second subclass.

3) Taken King subclasses;

Hunter - Nightstalker
Warlock - Stormcaller
Titan - Sunbreaker

These subclasses were introduced in Destiny with the Taken King expansion pack. They are earned by the exact same method as the second subclass, but you must be level 15 for the artifact to drop in patrol. EDIT: I never got these to drop before level 15, but several people have reported getting them at level 14.

[Note]: I am a warlock, and have been playing with a titan and a hunter. We all had exactly the same requirements for unlocking the subclasses, so there is no difference based on your class. These are however our first characters, so it COULD be slightly different for your second or third. If so, simply follow the instructions in the milestone you are given upon collecting your artifact from a chest.

Just a quick one -- I gathered all three subclasses and the achievement didn't pop.

Logged out (via Settings), back in and it then popped. Don't panic!

For Hunter classes are

Arcstrider (start with)
Gunslinger (Level 7+)
Nightstalker (Level15+)

Then open chests you find anywhere it seems, and you'll get a broken relic in with you must kill enemies until you hit 100%. This includes missions or cruicible.

Then travel to Earth, start "Shard of the Traveler".
Proceed to teleport to the Dark Forest.
Kill some enemies.
Teleport yet again.
Kill enemies, then intertact with the shard.
Use your new ability against enemies.

This is the case for the 3rd and final subclass as well.

*Heroic public events charge this faster. But killing anything charges this.

UPDATED FOR POST WITCH QUEEN:

When you first create a character now there is a tutorial new lights have to follow. Go to Shaw Han and complete the new light questline. Once you do a little tutorial you are tasksed to go to the tower. Find Ikora in the second part of the tower past the weapons vendor Banshee and the engram decoder Rahool on your right as you spawn in. There you can acquire two quests to unlock your other light sub-classes. Once you complete those requirements and turn in the quests your other two sub-classes should be unlocked and the achievement will unlock. If you used a level skip and have all sub-classes unlocked but no achievement, then equip each of the sub-classes until the achievement unlocks (this can be a little buggy with the skip).

If you are here then you are playing post forsaken and have boosted a character to lvl 30 skipping the missions to get your other sub classes.
DO NOT PANIC
It may seem as if the achievement is locked but it is not.

You will receive a mission from Ikora that will require you to get feathers of light from high leveled enemies. My suggestion is to do public events on the tangled shore because the enemies will drop then quite frequently.
Once you get 200 feathers(do not worry its not as hard as it looks) a mission on IO will appear.
Towards the end of the mission you will receive a seed of light to use on your character. It does not matter which sub class you choose just go to one of them and use the seed. Once you do that subclass will start to become unlocked. Once you kill all the enemies and finish the mission the achievement will unlock once you go back to orbit.

I had this issue happen to my warlock and it unlocked once i finished the mission.
Hope this helps any panics you may of had and happy hunting!

This is more of an add-on to the above solutions. If you do Heroic Public Events once you find the Relic, you will recharge it a lot faster. For the Hunter arrow, I did two normal events and was at 8%. Then I did the Heroic Event on Titan at the Rig landing zone and I jumped up to 19%. Please note, the Heroic Event at this LZ is on the inside against the Hive, not the one outside agains the Fallen.

Note 2: You can also recharge the Relic by killing random bad guys, it just takes really long. I was waiting for another Public Event to start and was sitting at 98%. Started killing random baddies in another area and it unlocked.