UPDATE: As far as I can tell, you can get this achievement with mixing up different classes gear; I popped the achievement with a set of DC gear in my GWF's inventory. Also, different classes can purchase the gear, you just need all the gear at some point on the same character. Doesn't have to be all at once!
Pilgrim's Gear is the middle level of the 5 new levels of gear in Chult. It is item level 460.
Pilgrim's Gear binds on pickup to account.
You can get Pilgrim's Gear by:
1. The Chest and Arms pieces in the Tarmalune Trade Bar Store. Chest is 80 TBs, Arm is 65 TBs.
2. The Foot and Head pieces one of two ways:
...A. Buying the appropriate packs off the Auction House, the Pilgrim Helm Pack or the Pilgrim Footwear Pack. At time of writing this there are lots of Helm Packs going for around 60k AD. There's only 1 Footwear pack going for nearly 900k AD, probably a price gouger.
...B. Buying the Zen Market item that includes the Head or Footwear Pilgrim Pack:
......1. Enchanted Keys and Pilgrim Helm for 2,250 Zen.
......2. Chultan Hunter Bundle, which includes the Chultan Hunter(increases by 10% drop of Chultan Riches and Forgotten Totems), a Lesser Bonding Runestone, and the Pilgrim Footwear Pack. That all for 2,500 Zen.
The Pilgrim's gear like the Huntsman's gear has more Recovery for my GWF than the other sets offer, otherwise a decent mix of the usual good stats.
Pilgrim's Gear has a mix of all-round useful bonuses:
Helm: +1% Damage in all of Chult.
Chest: Gain 100 Defense for each enemy you are engaged in battle with.
Arms: Gain 100 Power for each enemy you are engaged in battle with.
Feet: Wanderer's Vigor: When not in a party, you regenerate 3% of your Maximum Hit Points, up to 3000, every 3 seconds.