You'll land right near a save point and Shadow, who will join the party if you speak to him. (You can continue as a party of three if you want to, though there's no advantage in doing so.) The Floating Continent is a large area with a unique Encounter setup. There are four different Encounter tables, and each battle will choose one at random. As a result, the battles here are more varied than they've been in any previous location. You will gain crazy numbers of Magic Points from these Encounters, so it's a great place to train up new Spells.
Monsters on the Floating Continent
Seven different Monster types populate the random Encounters on the Floating Continent. They are ordered here from most to least common. The most common Monster, appearing in nearly half of all battles, is the Apokryphos. They have a powerful basic attack and a Silencer special that inflicts Mute. This makes it tougher to exploit their weaknesses to Thunder and Water (they are also vulnerable to Pearl attacks such as AuraBolt). Physical attacks work well, but they have 1,900 HP and won't go down easy. If Strago is in the party and the party's levels work out, you can learn L.3 Muddle, L.4 Flare, and L.5 Doom from an Apokryphos. They will use one of them at random as a counterattack when they are the last remaining enemy. Unlike the usual iterations of these Lores, the Apokryphos will only target the attacker with them, so use this to your advantage.
A third of all battles will feature Misfits, who are always in the company of other monsters (often Apokryphoses). They are undead, so don't have Shadow or Locke use the Assassin against them. They look and act similarly to Ings, though they use Lifeshaver even more frequently and don't have any counterattacks. You do have a short period after their second attack to hit them before they have a chance at using Lifeshaver, and this is the time to exploit their weakness to Fire. If you have a powerful mage, Fire 2 could potentially take out a whole group of them. Gaia Gear will negate Lifeshaver as well, making Misfits trivial to kill.
Brainpans are also undead, and have two dangerous special attacks. They will sometimes use Smirk to Stop one of your Characters. The Remedy or Dispel Spells will remove Stop. When there's one Brainpan all alone, it may use Blow Fish to deal exactly 1,000 damage. This is obviously a lot of damage to take at this point in the game, but Strago can learn Blow Fish if he's in the party. (Note that Strago cannot learn Blow Fish if he is Stopped when the Brainpan uses it.) Brainpans have the usual undead weaknesses, including Fire, and they are additionally vulnerable to Thunder.
Behemoths are big, powerful, and have 5,800 HP. Their basic attack is already nasty, and Take Down deals three times as much damage. A sole Behemoth won't even bother with normal attacks, using Take Down exclusively. Don't use summons against them, as they will counter with a powerful Meteo attack. Other Spells are fine, and Ice 2 in particular will help you cut through their massive HP totals.
You've seen a Ninja before, but here they always come in pairs. Thunder is your best bet against them, especially if they counter basic attacks with Inviz to become Clear. Their skean attacks will accumulate quickly and weaken the whole party. They are sometimes accompanied by a Wirey Drgn, who can also appear as a trio. They resist physical attacks and their own physical attacks hit very hard, particularly Wing. If the last Monster on the battlefield is a Wirey Drgn, it may use the devastating Cyclonic attack. They have 2,802 HP and no exploitable weaknesses, but are fortunately somewhat rare.
Finally, there are Dragons, which only appear by themselves. They have the most HP (7,000) out of any of the basic monsters here, and may counter any given attack by Sneezing the attacker out of battle. They mostly use normal attacks, but these are the most powerful normal attacks you've seen to this point. Their alternate attacks cycle between Revenge (dangerous late in the fight), Tail (which will most likely kill the target), Blizzard, and Cold Dust (which inflicts Freeze). None of these is good news, and cutting through all that HP without the party all being Sneezed away is tough. They are vulnerable to Thunder, so hit them with all the Bolt 2 and Bolt Edges you've got. If you're really daring, you can try to steal a rare Genji Glove from them.
Traversing the Floating Continent
The Floating Continent has a strange new tileset and some paths don't appear until you walk toward them, so it's a confusing place to navigate. You'll get a quick example of this when you approach the stairs that seem to lead nowhere. Keep following the path, twisting around toward the bottom where the next area opens up. Heading back north, you'll see one of the blue orbs that act as chests on the Floating Continent. It contains the Murasame, which has a higher Bat.Pwr than the Tempest but lacks its special abilities and so may not be preferable for Cyan, if he's in the party.
The odd-looking plateau near the blue orb is a passage to another part of the area, but for now continue east. You'll see another blue orb, and a path to it opens as you draw near. Your prize is the Hardened, a new Weapon for Shadow, but to obtain it you'll need to defeat Gigantos. He opens battle with three Throat Jabs, each a devastating physical attack. He'll counter attacks with a combination of two normal attacks and another Throat Jabs. Thankfully, his normal routine is just to perform normal attacks, which are manageable. Since all of the major damage is predictable, you can heal up between attacks and plan for them. Gigantos is weak against Poison, so a Bio Spell from a powerful spellcaster is highly effective. The Gigantos essentially has no Defense or Mag.Def, so go to town with your best attacks.
Return to the plateau and dive in to be taken southwest. As you head south, you'll notice a switch on the ground. Press that to open the way further east. When the path forks, take the stairs down to the southern path. Don't take the second set of stairs, but instead head east to another transport plateau. This brings you to a central area with three transport plateaus. The western one leads onward, while the eastern one leads to a dead end.
You'll see a switch in the next area. This lowers a portion of the terrain in a way that opens a new path in a previous section. Another switch to the northwest opens a path back to that section. Proceed west past the newly opened path, then down two stairways. You'll find the path you created to the east from the bottom level. This leads to a long, narrow path that will eventually turn north. You'll see the other side of the dead end from earlier and a nearby switch. This switch reveals a transport plateau to the east, but before entering it you'll want to check out the one to the south—it contains a much-needed save point. You'll also find a Beret in a blue orb to the northeast.
When you're ready, head to the northern transport plateau. At the bottom of the stairs at the other end, you'll be given the option to return to the Blackjack. If you do, you'll need to start over from the northwest corner of the Floating Continent, though you do not need to deal with the Imperial Air Force again. You'll also leave Shadow behind, but can re-recruit him in the same place as before. Otherwise, head east to reveal a passage through the wall. Make sure all of your affairs are in order, because as soon as you begin the next battle, you start a sequence of events that will forever transform the World of Balance into the World of Ruin.