This is some information I've put together for those looking to start Zilart missions. Good luck
ZM1: After defeating the Shadow Lord, go to Norg for the cutscene.
ZM2: Talk to Gilgamesh in Norg.
ZM3: Talk to the Mithra leader in Kazham for the Sacrificial Chamber Key.
ZM4*: 3 Tonberry BCNM fight in Sacrificial Chamber. You will need 4 unlit lanterns (1 for each member), dropped by Tonberry Maledictors in the Temple.
ZM5*: Collect 7 Prays from Cermet Headstones. 4 of them will pop a NM. (Locations: La Theine [through Ordelles], Western Altepa Desert [through Quicksands], Cloister of Frost [through Fei'Yin], Behemoth Dominion (NM), Sanctuary of Zi'tah (NM), Cape Terrigan (NM), and Yuhtunga JUngle [through Ifrit's Cauldron](NM))
ZM6*: 3 Antican BCNM fight in Chamber of Oracles.
ZM7*: Stellar Fulcrum BCNM fight (top of Delkfutt's Tower) vs spoiler! (Recommended levels: 70+)
ZM8: Go to Norg and talk to Gilgamesh.
ZM9: Examine the gate in Hall of the Gods for a quick cutscene.
ZM10: Return to Norg and talk to Gilgamesh again.
ZM11: Talk to the Mithra NPC in Rabao.
ZM12*: Pot NM in Quicksand Caves. (Instead of taking the western drop to get to Chamber of Oracles, take the eastern drop and you'll see the ??? in the area you drop).
ZM13: Cutscene in Hall of the Gods. You can now go to Tu'Lia!
ZM14*: Ark Angel BCNM battles x 5.
ZM15: Not yet completed.
ZM16: Nope, not yet.
ZM17: Maybe some day
Note: ZMs marked with "*" will require a party.