If you only want general pointers in the right direction, use the MISSION GEMSTONE in-game command.
Step 1
Go to the Hinterwilds and defeat the wyrm in a multi-phase boss battle. Getting credit requires that the group ultimately defeats the wyrm and that you contributed at least some amount to the battle. (The exact amount of necessary contribution is unknown, but seems to be generous; as an example, a monk who landed only a few jabs of 5-15 damage each and a single Web got credit.)
Starting the battle involves locating a gigas ballista whose general location is signaled by monsterbold messaging, killing the gigases guarding it, and firing the ballista into the wyrm as she flies overhead, bringing her down to the ground.
The first phase of the real battle is a conventional fight against the grounded wyrm. In the second phase, the wyrm can take to the skies at times, preventing many grounded attacks from hitting her. Archery, magic, and other ranged attacks still work. In the third phase, the wyrm pulls back to Wyrmreach. Up to five characters who were alive and battling against her when she withdrew go along with her for the climax of the encounter.
The third phase can be the final phase if the five characters who went to Wyrmreach kill her. However, if they die in battle or are otherwise unable to kill the wyrm after several minutes, the wyrm will return to the spot from which she departed at the end of the second phase. At that point, if a remainder of the group is still there, they can kill the wyrm instead.
After defeating the wyrm, speak with Ulvrig in the Hinterwilds' Eldurhaart at the Altar of the Dragon (room 7503383 or Lich 30026, go door). He'll provide backstory, information, and access to offerings inside a chest, from which you'll get a cloth bundle containing an ancient crumbling gemstone and a tattered parchment note pointing you to Sylvarraend.
Step 2
Go to Sylvarraend and speak with Ilyra inside her cottage (room 13050013 or Lich 683, go cottage). Give her the note and the gemstone to get your next lead, pointing you to Zul Logoth.
Step 3
Bring 2010 silvers or so to Zul Logoth, 2000 being for cart travel and 10 for alcohol, and locate the troubadour. (He's a wandering NPC, so there's no specific room; just look around.) Ask him about Beylin, but he'll want a drink before he provides information; buy Hollow Leg Bock from the Bluerock Brewery (room 13014007 or Lich 16836, order 6 on the menu) and bring it to him, then ask about Beylin again. Now he'll want stitchings and direct you to Solhaven.
Step 4
Go to the Sanctum of Scales (accessible through Solhaven) and hunt patchwork flesh monstrosities until you get three stitchings by looting them. The stitchings you find are monsterbold messages, not physical objects.
The messages are pretty lengthy and hard to miss, but if you've been hunting for a while and haven't seen them, you can check MISSION GEMSTONE to see if you've already found the needed stitchings without realizing.
Step 5
Return to Zul Logoth and speak with the troubadour for your next lead, pointing you to Teras Isle.
Step 6
Bring 250k silver to the Golden Helm on Teras Isle (room 3003006 or Lich 1842) and locate the hound somewhere inside. (If the hound is nowhere to be seen, eventually it'll return, so just keep looking around the building.) Ask it about Beylin. Beylin will then arrive, so speak with him and ask him about purification. Give him 250k and the gemstone to have it purified.
Step 7
Return to Sylvarraend and speak with Ilyra about the next step with the gemstone, which needs a setting.
Step 8
Go to Moonsedge Castle (near Icemule) and speak with the whitesmith in the jeweler (room 4577057 or Lich 32431, go door) about creating the setting for your gemstone. He'll request that you hunt ashen patrician vampires.
Hunt ashen patrician vampires in Moonsedge until you get a message indicating that you've done enough. This message is comparatively easier to miss than the monstrosities from step 5 since it's shorter and there's only one message. If you've been hunting for a while and haven't seen anything due to screen scroll, you can check MISSION GEMSTONE to see if you've finished without realizing.
When done, return to the whitesmith and speak with him to create the setting for your gemstone.
Step 9
Return to Sylvarraend and speak with Ilyra for your final lead.
Step 10
Return to the Hinterwilds and defeat the sybil in a boss battle. The sybil can be accessed by touching the altar at the western dead end in the Pits of the Dead, then entering the archway. The sybil is a heavily magical-oriented boss with a wide variety of offensive magic, buffs, and debuffs. She can also summon archways in adjacent rooms that need to be dispelled (effects like Spell Cleave also work) since they shoot acid through the area.
After defeating the sybil and before leaving the room, get out your gemstone from step 2 and INFUSE it. This officially completes the quest and grants your first Gemstone!