Once you have control of the Falcon, every remaining area in the game other than Kefka's Tower is technically optional. You're not going to have much luck against Kefka with just three or four Characters, though. Your goal for the rest of the game is to gather your strength, recruiting old and new Characters, finding powerful Espers and equipment, and raising your levels. The game gently pushes you towards a few specific goals (such as by the bird flying to Maranda after you first obtain the Falcon), but the order in which you complete them is totally under your control.
Not every location or reward is immediately accessible, even with the Falcon. Sometimes you'll need to recruit a certain Character to access a location, or need a certain Spell to be able to actually finish a dungeon. These requirements are listed in the table below, so you know what quests are available to you and how to access the ones that currently aren't. This table includes a few sidequests and rewards you may have already earned, just in case you missed them the first time.
The table is listed in the same general order as this walkthrough. This order is based on relative monster level, in-game hints, and some inferences about what order the game is designed to be played in. It is not necessarily the best or easiest order to complete the game in, nor the order I would necessarily choose for maximum efficiency. It also doesn't include towns or other locations without a specific material reward, though the walkthrough will mention these. Likewise, this table doesn't outline locations that are interesting for other reasons, such as efficient leveling. This is just a list of where to get stuff.
If you want to avoid spoiling everything too far ahead, you can skip to the Next Page. Everything listed in this table will be detailed in the walkthrough to come.