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The Levistone is inside the Cavern of Ice, and you need it to raise the Airship. Sail north from Crescent Lake into the bay and dock at the port. Take the Canoe west along the river, then follow the forks north until you reach land. Level 28-30 is comfortable for the full clear.
Cavern of Ice
This dungeon loops between three floors multiple times. Cracked ice floors collapse when you step on them, dropping you to the floor below. Some falls are required to reach treasure and the Levistone itself. Spiky ice patches work like Mount Gulg's lava, dealing 1 HP per step to your entire party. Rest at the Crescent Lake Inn before entering; there's no way to heal inside.
Most enemies here fall into two groups. Undead (Specters, Wraiths, Mummies, Wights, Bloodbones, Ghasts) all burn to Fire and Dia as usual. Ice types (White Dragons, Winter Wolves, Ice Gigas) are weak to Fire; White Dragons also take extra damage from Thunder. Dark Wizards have no elemental weakness and can cast Death and other nasty spells, so kill them fast. Cockatrices still petrify, so keep Gold Needles on hand. Remorazz is a rare encounter here; you'll need it for the bestiary but can also find it on the way to the Citadel of Trials later.
Cavern of Ice directions
B1 (entry). No treasure. Follow the path east to the stairs.
B2. Head southwest and take the stairs down to B3. No treasure on this pass. B3's stairs bring you back up to a different section of B2 with the Levistone room. Walk around the path and enter the room. Avoid the cracks in the floor and grab both chests: Flame Sword on the left (approach from behind), Clothes on the right (guarded by 4 Dark Wizards). Swap the Flame Sword onto your Warrior; it replaces the Ice Brand you've been using since Mt. Gulg. Once both chests are open, step on a crack to drop to B3.
B3. Undead ambush you immediately. Dia or Firaga clears them. Head west across the spiky ice (1 HP per step) into the northwest room. Left chest: Mythril Gloves. Right chest: Ice Armor, guarded by 2 White Dragons. Put the Ice Armor on your Warrior. Walk south across the full length of the floor to a room with 6 Gil chests (over 41,000 Gil total; two are guarded). Take the stairs on the east side back up to B1.
B1 (second section). Ether in a chest immediately south of the stairs. Small room nearby has 10,000 Gil. Head north, take a right, then go south into a long room with 3 chests and a crack in the floor. Grab 9,500 Gil, a Sleeping Bag, and the Ice Shield. Equip the shield on your Warrior for ice resistance. And then fall through the crack to land back in B2.
B2 (Levistone). Avoid all cracks this time. Walk carefully to the Levistone and grab it.
Evil Eye
Evil Eye has 162 HP and carries three instant-death spells: Death, Glare, and Rub. Speed is everything. Skip defensive buffs entirely. But Haste your Warrior, and have every party member attack or cast Firaga/Thundaga. The fight ends in 1-2 turns if you don't hold back; any longer and the instant-death spells will start landing.

Pick up the Levistone. To exit, fall through one of the cracks. You'll land back on B3 and have to fight through the undead again. Retrace the B3 path east to the stairs. Back on B1, head north past the long room where you grabbed the Ice Shield. This time skip the crack and take the stairs northeast of that room to leave the dungeon.
Raising the Airship
Head back to Crescent Lake, rest at the Inn, and restock. Take the canoe west but head south this time, following the river until you reach land. Step off and walk into the small desert south of town. The Levistone raises the Airship from beneath the dunes.

You can now fly anywhere on the world map. Head northwest to the Citadel of Trials, where your party upgrades to advanced classes. Stock up on healing items before heading there; the Citadel has no save points.