The Fire Cavern is immediately visible as soon as you exit Balamb Garden. It can be seen beyond the forests to the east. Use the select button to pull up the map to orient yourself with the surrounding location. You should consider levelling up your characters a bit though before you make your way over. Make sure to draw tons of Magic as well and to junction your magic properly. Click on the Magic Junction Guide link for more information on how to do this.
Make your way down to the beach and you will encounter monsters called Fastitocalon (pictured above). They provide the most AP of all of the monsters in this first part of the world, so it is best to level up fighting these monsters specifically. They are weak to thunder attacks, which is perfect considering Quetzalcoatl does thunder-type damage. I would suggest having Shiva learn the Str-J ability as this will allow you to junction magic to your attacks. In the Fire Cavern, for example, you could junction Blizzard to your attack to cause extra damage.
Also make sure to track down a Caterchipillar so you can draw out and stock up on some Cure magic. With some Cure magic on hand you will not have to go blowing through all of your potions when you venture into the cave. Draw up on as much of the other magic as you can as well – junctioning the various magic spells (with as many stocked up as you can) is a great way to build up the stats of your characters.
Head to the Fire Cavern when you are ready. Quistis will attempt to explain magic junctioning but I have a better explanation of the system in the Magic Junction Guide. When you are asked what time frame you expect to complete the challenge in select “10 min”. This will be more than enough time. The time you select will affect the SeeD ranking you receive at the end of the battle – you will receive a higher ranking for finishing the Fire Cavern faster.
The only thing that can really slow you down is all the pointless banter between Quistis and Squall. Skip past it and follow the path all the way to the boss. Other than a draw point for Fire magic there isn’t anything else to see along the pathway.
Boss Battle: Ifrit
Ifrit is the first boss, he’s the easiest boss, and he’s also your next Guardian Force. There are two main strategies to keep in mind:
1) Do not use fire
2) Use Ice (summon Shiva, use Ice magic or junction Ice magic to your weapon)
If you find either of your party members running low on health you can draw Cure magic directly from Ifrit (which will prevent you from having to waste potions or your own Cure magic).
Exit the cave once Ifrit has been defeated and make your way back to Balamb Garden.