ZM3 - Kazham's Chieftainess  

This is known as Zilart Mission 3.

I'm unclear as to the origins of the Zilart Mission numbering scheme, but in essence many of them are mere cutscenes for you to follow, with no rewards except for the loot you collect along the way (you get the Opo-Opo Necklace during ZM5, for example) and an advancement of the super-cool storyline. Talking to Kazham's Chieftaness sends you to Uggalepih Temple for Zilart Mission 4, the first one that involves any fighting. Quick summary (MILD SPOILERS):

ZM1: Travel to Norg, see cutscene, get map of Norg
ZM2: Talk to Gilgamesh in Norg
ZM3: Talk to Kazham Chieftaness
ZM4: Get 4 unlit tonberry lanterns, Uggalepih key, fight way into Den of Rancor, kill 3 tonberry NMs
ZM5: Collect 7 prayers from headstones around the world. Some are guarded by NMs.
ZM6: BCNM fight in Chamber of Oracles. 3 Antica NMs
ZM7: Cutscene that follows ZM6
ZM8: BCNM fight in Stellar Fulcrum (major boss fight)
ZM9: Talk to Aldo, then Gligamesh
ZM10: Examine Cermet gate in Hall of the Gods
ZM11: Talk to Gilgamesh again
ZM12: Talk to mithra in Rabao. Form alliance, kill a nasty NM in Quicksand Caves. This is the famous 'Mithra and the Crystal' Zilart Mission
ZM13: Enter Tu'Lia (congratulations!)
ZM14: Ark Angel Fight. One at a time or all 5 at once, your choice.
ZM15: Talk to Gilgamesh (again) and then head for Ru'Avitau Shrine (home of Kirin ^^)
ZM16: Work your way to the Celestial Nexus. Not as easy as it sounds. Even 75 players do this with sneak/invis the whole way.
ZM17: Final boss battle, and the end of the Zilart Missions.