Destiny 2's Forsaken expansion gave each class three new subclass skill trees with which to bother the game's long-suffering aliens. Compared to the original subclass trees, the new ones are more flavorful and synergistic, plus each has a powerful new super move. So you're going to want to unlock them ASAP.
How to unlock your first subclass tree
To unlock your first subclass, simply play through the campaign in Forsaken. After you clear the first mission, enemies will start to drop feather-like items called Visions of Light. These appear as little lights on the ground. Be sure to pick them up. Once you collect 100, you'll unlock a mission on Io which rewards a Seed of Light that you can use to unlock one of the new subclasses. Pick the class you want to play the most, because you won't get another new tree for a while.
Tougher enemies drop more Visions of Light, so to speed up this grind, look for public events in the Tangled Shore as you play through the story. These are filled with orange- and yellow-bar enemies which drop lots of Visions. Once you collect 100 Visions, go complete the Io mission as soon as possible so you can level your new subclass as you play through the rest of the campaign. You only get the new super at first, but as you deal damage with your super and grenade, you'll get access to your tree's melee and passive abilities too.
How to unlock your second subclass tree
You won't find more Visions of Light after you receive your first Seed of Light. However, you can get a second Seed, it just takes a little more effort. Start by finishing the Forsaken campaign, and after you beat the final mission, talk to Petra and Spider in the Tangled Shore to start the quest that unlocks the Dreaming City area. It's a straightforward quest, so just follow the in-game prompts and you'll bash it out fairly quickly.
Your target is the Blind Well activity in the Dreaming City. The Blind Well is split into four tiers based on difficulty. The tier-two, 540-light version of the Well can drop you another Seed of Light, so you'll need to replay it until you get one. You're going to need teammates for the Well, and you'll want to be at least light level 515 for the tier-two version.
You can start a tier-two Well by depositing the appropriate 'charge' in the center of the arena. If you don't have the right charge on hand, you can buy one from Petra in the Dreaming City using Dark Fragments, which can be obtained by completing other activities around the City. Failing that, find a teammate who has a tier-two charge. You can receive a Seed of Light from a round even if you don't deposit the charge yourself.
The Blind Well is a fairly simple horde mode not unlike Mars' Escalation Protocol. Once you start a round, enemies will spawn in waves. You need to kill enough enemies to fill the progress bar within each wave's time limit, but there are a few other things to keep track of. For starters, you need to stay within the bubble-like shields dotted around the Well in order to maintain the 'Touch of the Sky' buff which protects you from the 'Touch of the Deep' debuff. These bubbles move after each wave, so keep an eye out for the next one.
You'll also encounter glowing, seemingly invincible enemies in the Well called Servants of the Plague. To kill these Servants, you'll need to kill glowing enemies called Anathema. Anathema drop glowing orbs which give a buff called Harmony when picked up. Harmony allows you to damage the shields on Servants and also increases your super recharge rate considerably. Killing Anathema and grabbing Harmony orbs is especially important for the final round of the Well where shielded Abominations spawn alongside several Anathema. Deal with them in the same way as the Servants: kill the Anathema, grab their Harmony orbs, and break some shields.
Currently you can speed up your Blind Well grind considerably using a bug involving the exotic fusion rifle Telesto. Telesto shoots projectiles which explode after a short while, and for some reason the Blind Well registers these explosions as enemy kills. In other words, by firing a few volleys of Telesto you can max out the wave's progress gauge in an instant. This is a bug, mind, and one which will surely be squashed soon, but you can still use it while it lasts.
*Minor Last Wish raid spoilers ahead*
How to unlock your third subclass tree
Once you receive a Seed of Light from the Blind Well, you can't get another on that character. To get your third Seed and unlock your last new subclass, you'll need to play the new raid, Last Wish. You don't have to beat the raid, mind, since the Seed comes from the first encounter, but the raid itself is still level 550, so you'll need to power up as much as you can. Per this Twitch clip from Destiny streamer Datto, the Seed drops when you defeat the first boss, Kalli, the Corrupted:
We'll have more information on Kalli and the other Last Wish bosses in the coming weeks.