After you meet with Quistis, exit the Garden. This will take you to the World Map, where your entire party is represented by an oversized Squall running around the map. You can save your game at any time from the World Map. Out in the World, you will randomly come across monsters. In the vicinity right around the Garden, there are several different kinds of monsters:
Bite Bugs - Low health, low damage. Can draw Scan and Fire from them. Scan is a spell that allows you to see the health, level, resistances and weaknesses of enemy monsters when cast. Fire is a standard fire magic attack.
Caterchipillar - Medium health, medium damage. Will cast a Sleep effect. Can draw Thunder and Cure from them. Thunder is a standard Lightening Magic attack. Caterchipillars are found in the woods near Balamb Garden - but watch out, there are monsters in those woods you don't want to tangle with at this level.
Glacial Eye - High health, medium damage. Immune to Frost attacks, weak vs fire attacks. Can draw Blizzard, Cure, and Scan from them. Often accompanied by Bite Bugs.
I recommend you run around the map a little bit, engaging in battles with these monsters. The goal here is to stock around 100 of each spell you can get - Fire, Blizzard, Cure, and Scan. Spread your spells out across Squall and Quistis and don't forget to junction Magic.
After you've stocked up on a lot of magic, especially Blizzard, you can head to the Fire Cavern, which is to the "right" of Garden, near the woods. Quistis will ask to go over various things when you get there, like Gunblade usage and Magic Junctioning. You should already have Magic Junctioning down, but you may want to take the crash course on Gunblade usage.
Run up and speak to the Garden reps near the cavern entrance. They will ask you to choose a time limit. Note, if you fail to complete the Cavern in the selected time limit you will get "Game Over" and have to reload (so save from the World Map before entering). 10 Minutes is enough time, barely. 20 is plenty. However, to get the highest possible SeeD score (a little bit later in the game) you'll have to select the 10 minute timer, and attempt to use every second of it (the closer you get to the exact time, the better your SeeD score).