Is there anything I should do before the floating continent?
Plenty. :V Well, there’s not that much that is permanently missable, but there are preparations you can make that’ll make your life easier.
-Make sure you have at least one Ribbon available. Failing that, a Jewel Ring. You’ll need protection against petrification for a period of time where you only have a single character.
-Make sure Mog has the Water Rondo dance; it is unavailable later. To get it, either return to the Lete River, or take the Serpent Trench with Mog in the party.
-Purchase 99 Tonics and 99 Dried Meats. Not that this serves any real purpose but those items can’t be purchased later and so I like to carry a full supply, even if they’re mostly useless.
-Fight off an Intangir on Triangle Island. Have one character (with at least 1000 hp) cast Doom on it. That’ll make them available on the Veldt so that Gau can learn its Rage. Or kill lots of them, to quickly learn lots of magic, as they give 10 Magic Points apiece.
-Try and encounter a Grenade on the forested island left of the Veldt. It’s mostly Balloons, but a Grenade will show up every now and then, and fighting one there is the only way to make it appear on the Veldt, allowing Gau to acquire its Rage.
-Although the game’s difficulty level as a whole is extremely low, for a first-timer it’s a good idea to level up to at least 25, and have all characters learn all available magic. Pay special attention to Celes, Sabin, Edgar, and Setzer.
-Try and get Seraphim (in Tzen) Zoneseek, and Golem (Auction House, Jidoor). Although you can get them later on, having those espers and the spells they provide will make the following areas much easier.
Some tips for the Floating Continent itself:
-Equip Celes, but do not bring her along. You’ll need her later on, and won’t have the time to spare to do so.
-Bring Strago along. The place is chock full of Lores he can learn- L. 3 Muddle, L. 4 Flare, L. 5 Doom, Blow Fish, and Revenge come to mind, among others.
-Wait for Shadow. Make at least two attempts, choosing to wait both times, and he’ll come along and be available later on as a result.
Also, don’t forget to visit all of those storehouses that get unlocked for making a whirlwind tour of the Empire’s capital.