Destiny 2

As you progress through the story you will unite the 3 Vanguard members (Zavala, Cayde-6, and Ikora) and they will join all the other vendors at the Farm and will be told to speak to them.

Upon speaking to Cayde-6 you will be tasked with completing a milestone in the EDZ (Earth) to place down patrol beacons. Once the patrol beacons have been placed, you will be tasked with completing 3 patrols (just like in Destiny).

Once you've completed 3 patrols, return to Cayde-6 and you will unlock the milestone for completing public events in the EDZ. After completing 6 of these you'll be told to go speak with Cayde-6. After speaking with him the achievement will pop.

EDIT: As of the release of Beyond Light in November 2020, this achievement has been changed to simply require the completion of 5 Heroic Public Events. See guides for this similar achievement:



Up to date solution as of July 2020

You can do the Flashpoint wherever it appears; it's no longer limited to the locations listed in the description. I just popped this achievement by completing a Flashpoint on Mars.

Prerequisites

You must be at the soft power level cap. As of the current Season of Arrivals, this is 1000. This may increase in future seasons. I've seen some reports that once you reach the cap, you still won't be able to see the Flashpoint until the next weekly reset.

You need to complete the introduction quest for the world that currently has the Flashpoint (if that world has an introductory quest). These currently are:
- European Dead Zone (Earth): "Enhance!"
- Titan: "Enemy of My Enemy"
- Nessus: "Exodus Black"
- Io: "Dynasty"
- Mars: "Data Recovery"

There's a quest in the New Light section of the quest menu called "Introduction: Quests" that asks you to complete all of these, so that can help track your progress.

How to complete a Flashpoint

Every week, one world is assigned a Flashpoint. The current Flashpoint is listed on Bungie.net, or you can look at the Director and see which world has a gold challenge medal next to it.

Flashpoint progress appears as a challenge for that world's vendor. The following activities contribute to your progress:

- Lost Sectors: 10%. It appears to be 5% for killing the boss and 5% for opening the chest.
- Public Events: 15%. Note that only regular, clock-initiated public events seem to count. Player-initiated public events that have a permanent map marker don't.
- Heroic Public Events: 25%

After you reach 100%, speak to the vendor for a piece of powerful gear and your achievement. If it doesn't pop immediately (it didn't for me), using the switch profile option and reloading your character should make it pop.

Since Destiny 2 has gone free to play it looks as if the requirements to unlock Flashpoints has changed.

Originally these would be unlocked by progressing to a certain point in the main story, but as of New Light, you now need to reach 900 light with a character and this will then unlock the weekly challenges.

Flashpoints involve completing various activities (public events, lost sectors, adventures etc) on a target planet which changes weekly. Once enough of these have been completed you can then turn them in to the planet's vendor. To find out which planet is currently "hosting" a flashpoint, visit: https://www.bungie.net/en/Explore/Detail/Event/463010297