All information in this guide is correct as of Season 15, Season of the Lost. I have finished my time with Destiny 2 and will not be updating this guide further. If there are changes to this achievement with future updates it will be up to someone else to write that guide.
There are 19 different Collections tabs that keep track of your gear from various events. Unfortunately with Beyond Light a lot of the gear was removed from the game.
Solobrus22 earned this achievement using the European Collector tab which is for collecting gear on Europa. This means the achievement is not just tracking things that came out during the Forsaken expansion.
Steaksaucepie unlocked this achievement using the Exotics for Cruise of Osiris/Warmind
First, where do you find the Collections? While in game press the Menu button to bring up your inventory. Press LB twice to move over to the Collections menu. On the right side of the screen there are a list of panels under "Badges" those are what you are trying to complete. If you highlight one and press A it will open a menu that shows all the gear you are attempting to collect. You only need to collect the gear for one class, not all the classes.
Here are the ones that don't work:
Season of Arrivals - ignore this one unless you already bought the Season of Arrivals season pass.
Annual Pass: Jokers Wild - Gear has been removed from the game.
Annual Pass: Black Armory - Gear had been removed from the game.
Destinations: Dreaming City - the weapon Horror's Least is only obtainable from the Nightfall strike The Corrupted. If this strike is not in rotation this tab can not be completed.
Destinations: Forsaken - Tangled Shore gear (the top half) has been removed from the game.
Destinations: Curse of Osiris and Warmind - The planets Mercury and Mars have been removed from the game, thus gear is unobtainable.
Destinations: Red War - Planets Io and Titan have been removed from the game, thus gear is unobtainable.
Forged in Fire - Crucible gear for the collection tab has been removed from the game.
Playing for Keeps - Gambit gear from the collection tab has been removed from the game.
Now we can focus on the ones that we can complete:
Gear collected on Europa. To earn all the gear you will need to complete the Beyond Light story and the epilogue. You will need to do Sabotage missions to unlock the weekly Empire Hunt, then grind that until you get the Cloudstrike. Finally you will need to earn 1400 Triumph points in Europa triumphs.
Trials of Osiris
This one is a pretty small collection and might be a quick completion, but I still haven't unlocked Trials of Osiris so I can not say.
Gear collected on the Moon.
Exotics: Red War
There are 29 different exotics for this tab. They are all still in the game. 4 of them can be crafted from the Exotic Archive: Sturm, Rat King, MIDA Multi-Tool and Legend of Acrius. All the rest must be found as drops in game, or bought from Xur. One positive thing is that exotic engrams always give you something you don't own, so this will come eventually, after a long LONG time of grinding.
Exotics: Curse of Osiris and Warmind
There are 18 different exotics for this tab. 3 of them can be crafted in the Exotic Archive: Polaris Lance, Worldline Zero, and Sleeper Simulant. The rest must be found as drops in the game or bought from Xur.
There are 15 different exotics for this tab. Only 1 can be crafted: Ace of Spades. 2 require missions: Malfeasance and Wish-Ender. 1 is a random drop for completing the Last Wish raid. To complete this tab you will need a raid ready group and a lot of luck on your side.
Raid: Vault of Glass
Raid: Deep Stone Crypt
Sacred Duty: The Garden of Salvation raid
Raid: Last Wish
If you have a team going after these raid ones might be a pretty quick completion. That is assuming they don't remove any of the raids from the game.