Before continuing on your quest, you should take some time to familiarize yourself with the game's newest and most important advancement system: Espers. Each character (except Gogo and Umaro) can equip any one Esper at a time, and each Esper can only be assigned to one Character at a time. Equipping an Esper has three effects: it gives you access to that Esper's summon Ability, it allows you to learn Spells, and (for most Espers) it gives a stat boost upon advancing in level. Understanding how all three of these effects work and how to prioritize them is the key to maximizing your Esper usage.
Summon Abilities
Espers are the equivalent to earlier Final Fantasy games' summons, and indeed, you can summon the Esper you have equipped. This can only be done once per battle. These Abilities are similar to Spells except they tend to target all foes (or all allies) and often have effects you can't otherwise manage with normal Spells. This can take the form of a Statuses-inflicting spell that affects all enemies even though the equivalent Spell only has a single target, or even totally unique effects like that of Golem. Other Espers simply act as damage spells. These tend to cost more MP than they would otherwise, but can be immediately used by any Character that equips the Esper rather than needing to be learned first.
Of the three aspects of Espers, summons are generally the least important. They can allow all sorts of fun tricks, but they're rarely a primary part of your battle plan. They're just too MP-intensive and limited in use.
Learning Spells
When you think of Espers in FFVI, learning spells is usually the first thing that comes to mind. Every Esper teaches one or more spells, and usually at least three. Each spell has a learning rate, expressed as a multiplier. To learn a spell, you need to earn AP after battle. Each AP is multiplied by the learning rate for each spell, and when this adds up to 100, the spell is learned. From that point on, the Spell can be used by that Character. You can give that Esper to another Character and have them learn the same Spells, until everyone in the party knows every Spell.
In the vast majority of cases, Characters will learn Spells from Espers, but certain equipment can also teach spells (all of it only available in the World of Ruin). Most such equipment takes the form of shields—the various Elemental shields all teach their respective tier-2 attack spell at a rate of ×5, for instance. The Imp's Armor and other Imp equipment all teaches Imp at a rate of ×1. One relic, the Cursed Ring, also teaches a spell: X-Zone, at ×5. If you equip an Esper that teaches the same spell as a piece of equipment, the learning rate will be the sum of all the ways you are learning the spell.
It's important to note that you can interrupt spell learning and return to it later. You can see how far along a character is in learning any given spell on the Skills menu, under Magic. It's a perfectly valid strategy to teach a character the high-learning rate spells from an Esper, then switch that Esper to another character without first learning every available spell.
Also note that Terra and Celes learn spells automatically at certain levels. This has no effect if they've already learned a spell from an Esper, and will essentially set the learned percentage of that spell to 100% immediately regardless of its current value.
There are many potential strategies in determining who to teach what Spells in what order. You may want to give as many Characters as possible basic utility spells like Cure, or you may choose a few primary Magic users and prioritize teaching them the most powerful spells. It's up to you, but having a strategy will definitely help with planning. That said, don't focus on learning spells to the exclusion of all else. Level up bonuses are important as well.
Level Up Bonuses
Most Espers offer some kind of bonus on level up. These bonuses are usually applied to the four main Attributes (Vigor, Mag.Pwr, Stamina, and most rarely Speed) and take the form of a +1 or +2 to that stat. The exceptions are bonuses to HP and MP, which are expressed as +10% or +30%. These percentage bonuses apply to the amount of HP or MP that would have otherwise been gained at that level, not bonuses to the overall total.
These bonuses appear small, but they can add up in dramatic fashion. Character Attributes tend to have values in the 30's, and these never increase normally. Equipment can boost them, but only Espers yield permanent gains. Since all of the main Attributes can get as high as 255 (though Vigor effectively caps at 128), there's a lot of room to grow. The effects of increasing Attributes tend to be linear, meaning that gaining 15–20 levels with a +2 at each level can double the effectiveness of some attacks.
Concentrate on Vigor for physical attackers and Mag.Pwr for magic attackers. Stamina is the least useful Attribute to boost, though it does help prevent Wound effects. By far the most effective Attribute to increase is Speed, but only one Esper offers such an increase (and you may lose that Esper almost immediately). HP and especially MP are always good to have more of as well. Unlike other bonuses, these vary by level, so you may want to target them at certain times for maximum effect. (See the Level Chart for the base HP and MP gained per level.)
It's worth noting that Espers found later in the game tend to give better bonuses. Since the amount of times you can level up is limited, to truly maximize your Characters you'd want to level as little as possible. I would argue it's not worth giving up power during the game to gain unneeded power at the end of it, but min-maxers will want to keep this aspect of leveling in mind.
General Advice
In the long run, the most effective Esper strategy will prioritize level up bonuses. If you are willing to micromanage and make sure every characters gets the optimal bonus every time they level, your party will become absolutely unstoppable. However, the game really isn't difficult enough for that to be necessary. Still, keeping an eye on when Characters are nearing a level up is useful, as the level up bonus on an Esper is irrelevant at all other times. The best approach is a balanced one. Consider all the aspects of your Espers and assign and re-assign them as appropriate. Have a plan for each Character and execute on it, and you'll see the effects start to add up faster than you'd expect.