In the first cave, there is a secret path leading right on the elevated part of the first screen. It contains the only other Ribbon in the game that you can find (Morphing Master Tonberi in the Northern Craeter doesn't count.), so you might want to take some time and look around for it. The rest of this first cave isn't so hard, just push the boulder to remove the icicles, the go up and on. If you happen to run into the enemy Stilv here (and you will, eventually), try to manipulate him. He has two good Enemy Skills, Trine (which you might have gotten from the Materia Keeper in the Nibel Mountains, or from Godo in Yuffie's sub-quest) and Magic Breath, a nasty spell that can actually destroy a weaker party. Make sure at least one person has over 2000 HP to take the hit.
A few cliffs later, you'll find yourself in a second cave, with a save point and some good treasure. First you'll have to go right and around the bend. You can learn Bad Breath from the Molbors in the outside areas here, but be careful--have at least one person wearing a Ribbon to counteract all the bad stuff this spell gives you. You'll eventually wind up in a room with four huge icicles hanging from the ceiling. Approach one, and you'll have to fight it, along with some bats. Don't mess around with the bats, just go for the Icicle. When you win, the icicle will fall, and you'll be given the option to jump down. Don't. Instead, wait until all the icicles are gone, grab the treasure, and then jump down. You'll see that the fallen icicles have opened up a new path for you, and given you access to that treasure chest on the left.
Outside, the cliffs are getting trickier to navigate, so make sure you keep your temp up! When you finally make it indoors again, hit the save point and use the little free healing spring they provided for you (it's that white stuff in back there). Now, before you go on, take a minute and set up your materia and equipment for a battle against a boss. Take the most powerful stuff you have, because this next boss will prove challanging for any party. Twinhead has two independant heads (One uses Fire, the other Cold, and they can both use a devastating Earth attack.) When you finally do kill one of the heads, it hits all your people for 1500 damage as it dies. To avoid lots of pain and wasted time, try doing this. Hit him once with your new friend Alexander, then have everyone focus on one head (the one closer to the screen has a Protect Ring you might want to steal), then the other. Make sure to keep healing as much as possible, because if one of those heads dies and you're not ready for it, you might be in trouble. When you make it past Twinhead, go back and use the healing pool. If you want, you can hang around here and fight some Blue Dragons for lots of experiance. When your done with that, go on to the Crater and the Whirlwind Maze.