On November 10, 2020 this will no longer be the best way to earn the achievement, as many of the destinations will be removed from the game. I will update the solution once Beyond Light releases.
This may not be the fastest guide considering some may have every exotic.
If you pause the screen and tab over to the left to collections you'll be able to see all of your collections. In my humble opinion, the easier route for this without having to wait for Xur to have the exotic you need is Destinations: Red War.
For this, you need to visit the base game planetary vendors and earn/purchase the armor from each of them.
So look at it like this:
Devrim Kay on EDZ
Sloane on Titan
Failsafe on Nessus
Asser Mir (That was not a typo) on Io
I do not believe that you need Vanguard or Crucible Gear for this as it changes and the planetary vendor's stays the same.
If there's anything incorrect in this, feel free to let me know in the comments.
One thing I will add is that with the way planetary rep works now, you'll need the same planetary resource to level up with the vendor.
EDZ : Dusklight Shard
Titan: Alkane Dust
Nessus: Dattalattice (Not sure if spelled right)
A quick way to earn get these is from the Spider at the Tangled Shore. Keep in mind that his stock changes daily.
Another way to earn planetary mats is through bounties the vendors will sell. They cost a measly 250 Glimmer, and can range from chores like "Two Patrols on X", "30 Precision Kills", or "10 Grenade Kills."
EDIT 11/18/2018: One of the best ways to earn planetary mats, besides buying them or the vendor bounties, is through the Region chests. Each one gives you roughly 10 for the planetary material said chest was found on.