This is one of the most popular quests, since the reward is a great set of armor which can be upgraded to your specific profession later on (more on that at the end of the guide). The first stop is to get to any of the 3 Independent Professionals Society buildings in Jobe City. Walk up to your professional representative and after a bit of a chat you'll know that he or she will need insignias (and 15 million credits for a full set) to give you the armor pieces.
You will need 2 backpacks to store all the insignias, because you will be collecting a total of 30. The Enel and Sphere Insignias drop in Elysium, it shouldn't be much of a problem to get from mobs opposite to your side or those which are not from your own faction. The sames goes for the Ocra and Roch Insignias that drop in Scheol.
Note: I recommend you be at least level 100 in Elysium and level 125 in Scheol if you intend to solo this quest, if you have a few friends with you someone as low as level 80 could get this done in a short period of time.
The only real problem is getting insignias which belong to the opposite faction, because your choice are limited. You can attack a mob from your own side (which I strongly advise against), start looking for mobs which are not sided with the redeemed or unredeemed (which takes a bit longer to do), hope that players from the other side attack the mobs from your side and maybe be able to loot some insignias from their kills or hire someone to do it for you.
Alternative location for Scheol insignias is actually located in Elysium. There's a dungeon just a bit north of North Incarnator, where neutral mobs on second floor drop them. Drop rate isn't really good, but neither is in Scheol.
You can get those for free from your local tailor. Unfortunately, no tailor is stationed on Jobe islands, so you'll have to travel to Rubi-Ka to pick them up. Tailors are usually stationed in their own shops, marked as "Tailor" or "Boutique". There's one in Borealis and Newland, one in West Athens and a few in Omni cities (Entertainment and Trade host one, not sure about Rome). Miir shops unfortunately don't provide them.