To enter the Cyclone, simply approach it after obtaining the Wyvern. The Cyclone is positioned just northeast of Fynn. You can Teleport out if you need to, and the inn in Fynn is just a short walk away.
Cyclone 1F
There are a number of new Monsters on the 1F of the Cyclone. Vampire Girls, already annoying due to their Charm ability, are now accompanied by a considerably hardier Vampire Lady. Fortunately, her Charm 9, while more accurate, targets only a single Character. She can also use Sleep 16 on the party, and has over 1,000 HP, so she's no pushover. All of the other common Random Encounters contain only Monsters you've seen before: groups of Big Horns, a Hill Gigas in a new combination with Bombs and Captains, and large groups of Cockatrices.
More rarely, you'll fight a Wood Golem accompanied by up to two Royal Guards. All Golems have high Magic Resistance, but you can bypass that of the Wood Golem with Fire. They have six accurate attacks that can cause Sleep. The Royal Guards also cause a Status Ailment with basic attacks, in their case Poison. They have 870 HP but moderate Defense and no Elemental weaknesses, so physical attacks or non-Elemental Spells are your best bet. Another new monsters is the Lamia, a lesser version of the Lamia Queen. Like Vampire Girls, they use Charm 6 on the whole party. With 1,000 HP and no Elemental weaknesses, there's no particularly quick way to take them out. They are usually accompanied by Captains. The rarer encounters in the area are with a Devil's Bloom accompanied by up to two Gottoses, and a group of Royal Guards.
The layout of the Cyclone is surprisingly straightforward, as each floor consists of nothing but basic hallways to follow. What it does have is several branching stairways leading to treasure. As a general rule, if you take every stairway as you come to it, you won't miss anything inside the Cyclone. Since you cannot return after defeating the boss, you'll want to follow this rubric in order to get all of the valuable Treasure here.
The 1F is among the simplest. You'll start in the southwest and need to proceed east. The stairs there lead to the Ripper and Poison Axe on 2F and 3F, respectively. You'll have to either walk or Warp back to 1F to proceed. The stairs that lead to the Emperor are in the northeast, though you'll want to take the central path rather than the northern one to reach them.
Cyclone 2F
The Monsters on 2F follow the same basic pattern as most dungeons, with each encounter becoming one step more common. Gone are the Big Horns, and new are rare encounters with 1–2 Wood Golems unaccompanied by Royal Guards. The basic Wood Golem/Royal Guard encounter is now twice as common, so expect to see a lot of both Monsters
Most of 2F is on the Treasure path. The main path upwards is just a short hallway in the northwest. The eastern stairs open to a north-south path. To the north, you'll find the Ripper, an interesting and deceptive knife. It is a large upgrade over the Orichalcum for any knife user, but the damage it claims to deal in combat is 20 points higher per hit than its actual damage. Whether this is due to a bug or some strange joke is unclear, but the stats on the equipment screen indicate its actual power. With the Ripper in hand, proceed south and around to find the stairs to the 3F and the Poison Axe.
Cyclone 3F
Vampire Girls and Vampire Ladies will not appear on the 3F, but the similar (but tougher) Lamias are now a common encounter. Expect even more Wood Golems and Royal Guards. The new rare encounter is with Eyemoebas and Malboros Terra. The latter you've probably seen before, and Eyemoebas aren't as scary as they look. They absorb Poison and resist most Status Ailment Spells, but aside from six Poisonous attacks, they aren't a major threat. This is the kind of battle that will take some time but shouldn't drain too many resources.
The Treasure part of the floor is a basic hallway with some damage floors. The Poison Axe at the end of it is a huge upgrade for any axe user in the party. Aside from a large increase in Attack over the Ogrekiller, the Poison Axe both inflicts Poison and is Poison-[elements/ text=Elemental.
The main part of the floor is almost as simply laid out, but contains the second Treasure branch of the dungeon. The first stairs up you'll see, in the northwest corner, leads to the Diamond Gloves and Diamond Helm above. The southwest stairs lead onward to the rest of the dungeon.
Cyclone 4F
Once again, there is little change to the Monsters on this floor. You won't see another Hill Gigas or Bomb, but the rare encounter is with a group of dangerous Wizards. Unlike when you fought them in the Coliseum, you should be well-equipped to take them out quickly now with physical attacks or a solid Spell for each. They do still sport an array of dangerous and annoying Spells though, and with a bit of bad luck can wreak havoc on the party.
Most of the 4F is the northern Treasure area. The stairs are surrounded by a circular hallway, but you're interested in the northeast and northwest paths. The Diamond Helm is to the northeast, and the Diamond Gloves are to the northwest. The various pieces of Diamond Armor have the highest Defense you've seen so far find and resist Lightning, but at the cost of huge penalties to Evasion (and of course, Magic Accuracy). Use them wisely to improve your Defense, but not at the cost of other factors like stat boosts.
The main path starts in the southwest and proceeds east. The two staircases are nearly side-by-side. The first leads to the Defender and the Diamond Armor, while the second will get you closer to the climactic fight with the Emperor.
Cyclone 5F
Wizards are a lot more likely to appear on 5F than they were on 4F, though they're still not that common. The new rare encounter is with several of them accompanied by Captains. The Royal Guard battle without a Wood Golem is also common, though you won't find any more Cockatrices here.
The 5F has probably the most complicated (and deceptive) layout of any floor in the Cyclone. The main path is just a short hallway in the northeast, but the Treasure path splits in two in the middle of the floor. There is a long eastern route over damage floors that leads to the 6F, but there's nothing up there except four empty trap rooms. The real prize can be found further south, in the form of the Defender, an absolutely amazing sword and one of the best Weapons & Shields in the game. It's not that much more powerful than the Ice Brand, but it offers +8% Evasion per sword level, which can offset a whole lot of heavy Armor. This will be even more relevant in the near future.
There is a second chest on the path west from the Defender, which contains the Diamond Armor. It is guarded by one or two Generals, the latest in the line of armored soldier Monsters. With 100 Defense and 1,800 HP, they should be approached like bosses. This is even more urgent because of their 10(!) highly-accurate, 120(!!) Power attacks. Even Blink is going to have a hard time with that. Status Ailments Spells like Curse or even Toad may be your best bet here. They have a number of fantastic drops you might see if you're lucky, as well.
Cyclone 6F
Even more Wizards are, predictably, the Monsters to watch out for on the 6f. The new rare encounter combines Killer Mantises and Vampire Girls, which are a lesser threat than most encounters on this floor. Unfortunately, you do need to do a lot of walking on this floor, so you'll see several of these encounters no matter what you do.
The 6f is divided into an inner potion and an outer portion. The inner portion is reached via the Treasure path from the 5f, and consists of a door leading to a trap room in each cardinal direction. You can safely avoid this area. The main path winds around the inner area, and you can travel either north or west and reach your destination. The western route may be slightly faster since the one Treasure on the floor lies near the end of it.
This Treasure is the Wind Flute, an interesting Item that casts Cyclone 13, a non-Elemental ability that hits all Monsters. It's unlikely to do a whole lot to any individual Monster, and has a 20% break chance, but despite its limitations it is guarded by a boss-like Monster. The Green Dragon is similar to the White Dragon from the Mysidian Tower, and hits as hard as the Thunder Gigas. It's Poison Cloud 16 is probably the lesser threat. With 3,000 HP and 150 Defense, tactics like the Blood Sword may be useful. It has a weakness to Lightning that you can exploit as well.
Cyclone 7F
The Monsters on the 7f are the same as those on the 6f, boss fights notwithstanding. It's a very simple floor, as you can simply head north and approach the Emperor. Doing so will trigger a fight, but not with him. Instead, you'll fight some of his Royal Guards. The second fight is against the same group. Only after defeating two waves of Royal Guards will you be able to attack the Emperor, and even then, he has defenders.
You'll have to take out the Wood Golem before you can directly attack the Emperor, though you'll also have to defeat a pair of Royal Guards to win the day. The Emperor is a bit of a pushover, with middling Defense and only 1,290 HP. He can use Blink and Barrier to enhance his Defense, or use an Elixir to heal up, but he poses little offensive threat. If you thought the game would end here, the Emperor's anti-climactic stats and tiny battle sprite should hint that this is not the case.
When the Emperor is defeated, the Cyclone disappears and the party is automatically transported back to Castle Fynn.