The Snow Field
With the Snowcraft in hand, you can now traverse the Snow Field north of the mountains behind Salamand. To reach the entrance to the Snow Field, follow the mountain range west then north from Salamand. You'll automatically board the Snowcraft when you enter the white, snowy area.
The Snow Field only contains two types of enemies, Snowmen and Icicles. Snowmen are no big threat, on par with enemies you've seen earlier. Icicles can be a resource drain, though, because of their high Defense and HP. Fire will devastate them, but MP will be at a premium in the Snow Cavern so you may want to deal with them physically even though it will take longer. Upgrading to [Mythril] Weapons will help you pierce their Defense.
This is the first area where you will consistently find Rank 2 Monsters. In simplest terms, an increase in rank means that everything you use in combat will get +1 skill advancement per fight. It also indicates tougher monsters, at least in theory. In practical terms, rank increases have the largest impact on abilities you use once per combat (e.g., if you're casting a variety of Spells) or during short fights where you'll only attack once or twice. The longer the fight lasts, the less impact rank will have on your Weapon/Shield skill ups.
The Snow Cavern is a tough dungeon and you can't Teleport out, so if you want to build up your characters before delving it, the Snow Field is a good place to do that. When you're ready, cross the Snow Field eastward to find the entrance to the Snow Cavern.
Snow Cavern B1
There are all sorts of new enemies to be found in the Snow Cavern, and many of them are nasty. Dead Heads are similar to Soldiers, and shouldn't be a problem to take down physically at this point. The Ghouls you will find have decent HP but little defense. The danger from them comes from their ability to Paralyze you with their strong attacks. Shadows are similar foes, except their attacks inflict Darkness instead, which requires Esuna to remove. Grenades are much nastier versions of Balloons. If you can afford to use the MP, strong Fire or Thunder Spells can take whole groups down in one shot, preventing Self-Destruct from succeeding.
The path through B1 is very simple. At the first fork, you can follow the northwest path to find 100 gil. South of there, the path loops back to the east and leads to another chest, this one containing 150 gil. As you near the stairs you may notice a door in the wall. You cannot enter this door, but rather it serves as a convenient exit from B5. You'll need it, as the Teleport and Warp Spells do not function within the Snow Cavern! This will make backtracking to restore HP and MP much more difficult. You may have no choice but to fight your way through, conserving MP whenever possible. If you have Garlics dropped by Soldiers, they can be a useful alternative to Fire when confronted with groups of Undead.
Snow Cavern B2 and B3
The encounters on B2 are much the same as those on B1. The only new threat is a rare encounter with Dual Heads, but they are quite dangerous. They have a lot of HP and Defense, and no Elemental weaknesses, so it may be best to stick to physical attacks, with perhaps some support from a Blink Spell. These Giants are more common on B3
B2 is divided into two parts, and you'll spend little time here the first time you reach it. There is a door leading to a trap room near the stairs to the center of B3. The watery eastern path leads to a chest containing an Antarctic Wind, a powerful Ice-Elemental Item that can defeat the otherwise nasty Adamantoise you'll find at the bottom of the Snow Cavern in one shot. The western path from the central stairs splits, leading north to more treasures on B2 or south onward to B4.
The second part of B2 is a simple path with two short branches. The first branch hides a Battle Axe, while you'll find an Antidote in the second. At the end of the path, you'll be confronted by a team of Grenades when you open the chest. Defeat them to earn a Mythril Mace, a good Weapon if you have a staff-wielding non-mage. (Even if you don't, it may be worth training someone in the use of the staff in preparation for a powerful weapon you'll find in the next dungeon.)
Snow Cavern B4
The terrain gets a bit darker on B4. Fortunately this isn't particularly indicative of more dangerous Monsters. Dual Heads are still the main group to watch out for, and are more common than they were on the floors above, but there's nothing you haven't seen before waiting for you here.
The layout of the floor is more open than those above, with a watery central area and passages around it like spokes on a wheel. You'll start in the southwestern area, and the stairs to B5 can be found in the opposite corner. There are four chests to open on the other branches. To the northwest you'll find a Potion and just east of there, a Blizzard Tome. Like the Antarctic Wind found above, the Blizzard Tome can be used as an item to deal heavy damage to the Adamantoise. If you don't need to teach the spell to any Characters, and you want to hold on to the Antarctic Wind, this is a good alternative. Be aware that it won't finish the boss off by itself though, and you'll probably need to cast Blizzard yourself as well in order to win the fight.
The southern path hides a Potion, while in the southeast you'll find a powerful Mythril Spear. It's a nice upgrade for any spear user if you haven't already purchased one, and well worth using to pierce the high Defense of some Snow Cavern Monsters. Once you've picked up all the items you need, head east from the central room and then north. The door on the way to the stairs to B5 contains another trap room.
Snow Cavern B5
The bottom floor of the Snow Cavern is huge, consisting of two full-size areas filled with Monsters. You'll start off in the northeast corner with several paths available to you. Be careful while exploring, as basically all the Monsters you'll fight here are dangerous. The western path has a door to a trap room and a chest containing a Mythril Shield. Be sure to pick this up if you haven't purchased one.
If you head southwest from the stairs leading back to B4, you can follow the water into a large, open room. In the corner you'll find a chest guarded by Shadows and Ghouls. Defeating them earns you the interesting Ancient Sword. This weapon is more powerful than the Mythril Sword but painfully inaccurate. However, its power to inflict Curse on those you hit with it can be very useful, especially against bosses. Curse halves a Monster's Defense and Attack scores, both allowing you to break through and hurt them physically while protecting your front-line fighters. Its low Accuracy makes it a liability as a primary weapon, but it's well worth keeping in the inventory of someone with a few levels of sword skill.
When you're ready to continue, head through the door in the southeast corner of the floor. This leads to a room full of beavers, which Guy can apparently speak to. Speak to the one in the northwest corner about the [Goddess's Bell], and it will tell you that there is a path in the northeast corner of the room. This leads to the second half of B5.
There is a long and winding path to the Goddess's Bell and the Adamantoise guarding it, but it contains no branches. You'll find the same Monsters along the way as you have in the rest of B5. There is a Silver Cuirass in a chest near the boss. When you're ready, perhaps after healing up a bit, engage the Adamantoise in battle.
The Adamantoise has a frankly unfair Defense of 60. You can try Cursing it with the Ancient Sword, but even then it is difficult to deal much damage with physical attacks. Your best bet is to exploit its vulnerability to Ice. A leveled-up Blizzard Spell is devastating, as are the Antarctic Wind (which will kill the boss outright) and Blizzard Tome found elsewhere in the Snow Cavern. They may be your best bet if your black mage is out of MP at this point. The Adamantoise is a rank 5 Monster with an attack to match, so try to end the fight as quickly as you can. Blink can help anyone with a decent Evasion.
Once the boss is defeated, examine the wall where it was standing to find the Goddess's Bell. Even with it in hand though, your work is not yet done. The nearby stairs lead back to that one-way door on B1, but once you reach the long stairs leading to the cavern entrance, you'll see that your path is blocked by another boss, Borghen. He doesn't have any powerful defenses like the Adamantoise, so even with 240 HP you shouldn't have a problem taking him down. He can use Heal 3 on himself, but that will only delay the inevitable.
After the fight with Borghen, you may want to unequip Josef if he's carrying any valuable items. He will leave the party on the way out of the Snow Cavern. You'll have to return to Salamand as a trio.