The Flying Fortress is a five-floor dungeon floating above the Tower of Mirage. While this makes the trek to Tiamat a whopping eight floors long, most of the Flying Fortress floors are relatively straightforward, dedicating most of their space to large amounts of great Treasure (none of which is guarded). As such, don't hesitate to cast Warp2 and heal up if you're in trouble, since Tiamat (and possibly Warmech) await you on the top floor and it's not that long a trip back up.
The first two floors are the simplest, consisting of a central area with various branches containing rooms full of Treasure. The warp to the second floor is to the north, and the warp to the third floor is to the south once on the second floor. The third floor is slightly more complicated, including a large southern area that contains nothing of interest. The warp to the third floor is to the west.
The fourth floor is a strange one, as it consists of nothing but cris-crossing infinite hallways. The warp to the top floor is two sections east and two sections south from the start, but it's easy to lose your bearings when you get into Random Encounters. The final floor is a straight hall with Tiamat at the end. Watch out, as Warmech is a rare but incredibly deadly encounter on this floor.
Tiamat is basically the nastiest part of every Dragon you've faced rolled into one, and with 2,000 HP. She is well defended and has no elemental weakness. The standard strategy of Steel and Haste along with the Giant's Gloves will do well here, with Nul Spells (particularly NulAll) and Shld2 being your best bets defensively. Alternately, you can try a Poison-elemental attack like Break or the Venom Blade to exploit Tiamat's weakness. It's a cheap victory, but it beats eating away at 2,000 HP!
The Floating Castle is a five-floor dungeon floating above the Mirage Tower. While this makes the trek to TIAMAT a whopping eight floors long, most of the Floating Castle floors are relatively straightforward, dedicating most of their space to large amounts of great Treasure (none of which is guarded). As such, don't hesitate to cast EXIT and heal up if you're in trouble, since TIAMAT (and possibly WarMECH) await you on the top floor and it's not that long a trip back up.