Destiny 2

There are multiple ways to get exotics.

1. Campaign. During campaign, you are awarded exotics at certain points. You get a guaranteed 2 during the story, and one after completion. The one after completion is an engram so you could get unlucky and get duplicates.

2. Quests. After completing campaign, you will unlock quests that are blue. Do these to get a purple schematic thing, in which it gives you things to do. Do these and get exotic weapons. 1 on each planet it seems. *Io doesnt have one currently.

3. Public events. They have a chance of dropping exotic engrams, probably more so if you do the heroic versions of the events.

4. Turning in your tokens and planetary materials. Every time you level up with said person, you have a chance.

5. Play Crucible. You can get them for completing matches.

6. Xur. He comes on the weekends from Friday until Sunday morning. He sells exotics for legendary shards. Usually different each week.

*As stated by Maka91, you can go to your vault while in the social space, and count the number of exotics you have. However, this is only true if you have one character. It only shows your current character's armor, weapons is for everyone.

You must collect a total of 15 exotic weapons and armor pieces. The best way to see your progress is the "collection" tab of your vault at the tower. If you have multiple characters, unique armor will count to your total. There are a ton of ways to unlock exotics, here are the best ways to tackle this achievement:

1. Guaranteed Story Exotics (3+)

During the campaign, you're guaranteed to unlock 3 exotics. The first two are guaranteed to be unique, and you can choose between multiple options. The final exotic comes as an engram and will unlock a random item.

You can also start a new character to grab some of the exotics you skipped out on in the first two choices. This allows you to grab all 6 of the exotics from the first two choices if you're patient enough.

2. Milestone Quests (Luminous Engrams) (Up to 3)

Many of the Milestones that show up upon completion of the campaign will reward "Powerful Gear" that could result in an random exotic engram that will automatically be decrypted. You can only do these once, and duplicates are possible, but exotic chances are very high. You can receive exotics after completing:
- Cayde-6's Flashpoint Milestone
- Shaxx's Call to Arms Milestone
- Zavala's Nightfall Milestone

3. Exotic Quests (3)

After completing the entire campaign, you'll unlock 3 special quests that lead to special exotics. They appear as blue bannered quests. They are as follows:
- MIDA Multi Tool on Earth
- Rat King on Titan
- Sturm on Nessus

4. Random Drops (Public Events, Token Exchange, Raid, Crucible, Chests, etc)

Exotic Engrams can be randomly generated as you play Destiny 2. There are many ways to earn these random drops, and some generate more than others. First, I would recommend using the "Fireteam Medallion" available at Tess' shop in order to increase your odds at good loot whenever playing. Then, you can grind whatever activity you enjoy, like Heroic Public Events, Strikes or Crucible and there is a small chance that you will earn an exotic engram. You can then take it to Rahool and decrypt it into a random exotic. Personally, I would recommend Public Events as these have seemed to give me the most exotic engrams for my time.

5. Xur (This could could vary in Destiny 2; will update with more information)

Xur is a figure that appears every on Friday, and Saturday, starting on September 15. He is a vendor and will sell up to 4-5 exotics per week, although the armor will be spread amongst the 3 classes. At this moment, it is believed that the currency required are legendary shards which are acquired by dismantling legendary or higher gear, and as random drops from some vendors.

The gear that Xur sells is locked every week, so there is a chance that he will be selling gear that you already have though.

So after the New Light update on October 1st 2019, you no longer received exotics for completing the campaign

You can still do the exotic quest in order to progress for the achievement

Also, this achievement will pop once you get 15 exotics, it doesnt have to be "Red War exotics"
Mine pop at exactly 15 exotics and I have exotics from the Forsaken DLC and others exotics from other updates

If your achievement doesn't pop at the 15th exotic maybe you need to use the log in and out method

Cheers