Statistics in Kingdom Hearts and its Final Mix version refer to the player characters', and his party members', personal attributes such as level, Hit Points (HP), Magic Points (MP), Ability Points (AP, used to equip abilities), Strength, and Defense. All characters start with a base level for each stat, which then increases at specific level ups. Stats can be increased with accessories and even permanently with items like Power Up, Defense Up, and AP Up. Similarly new accessories in Final Mix also exist that will increase the amount of Experience (EXP) gained. Final Mix also added the EXP Zero ability, which stops everyone from gaining EXP.
Magic Points and the charge gauge
Magic Points are denoted as blue blocks on the ring around a character's icon. Each block represents one MP, and is used to cast Magic, and use summons, as well as certain active abilities (in case of the first two it also affects their strength). Attacking an enemy or picking up MP orbs restores MP through the MP charge gauge, an orange outer line inside the blue MP gauge. The amount of charge gained through hitting enemies depends on the enemy, and whether or not you have MP Haste equipped. Similarly MP Rage grants charge upon taking damage. Finishers and Counterattack hits also recover more charge.
To recover one MP you must fill the charge gauge by one unit more than your current MP, so if you have 3 MP then you need to fill the gauge by four units to recover a 4th Magic Point. Because of this it is more beneficial to stay at a lower MP level throughout fights. Cheap spells like Fire and Blizzard also only cost one charge gauge unit, rather than one full MP.
Choosing the Dream Rod at the start of the game is the only way to actually increase your maximum MP, as those who picked it end up with two more than other players near the end of the game.
Leveling up and stat growth
Leveling up is done through collecting Experience, which can only be gained through defeating Heartless and other enemies (not counting Tech Points). The amount of Experience depends on the enemies' level and overall strength. Experience is also shared across all characters, whether they are in the party or not. Each time anyone levels they gain an increase to a specific stat, such as HP (+3), MP (+1), and Str, Def, and AP (all +2). The actual total end result for everyone varies, but Sora specifically gains the most balanced amount. Other stats that can increase upon leveling are the maximum number of item and accessory slots.
Other than level-ups, one other important moment that affects your personal stats is the weapon chosen and discarded at the start of the game in Dive to the Heart. Although the minimum and maximum stats are largely irrelevant, as they are nearly the same and due to the stat Up items easily equalized, they do affect max amount of the the otherwise unchangeable fixed MP and item slots. Although the choice does not affect normal stat growth per level, the chosen weapon does affect when you gain specific bonuses such as your third accessory slot, and your bonus Magic Points.
Lastly Final Mix made several changes in the maximum stats (and stats gained, presumably) for Sora, with a few minor adjustments to all other characters. For any party members the main change is a raise or reduction in their Ability Points, whereas for Sora he gains a lot more AP (to accommodate his new extra abilities), and loses out on some Strength and Defense (which can be easily mitigated).
Tech Points also count as Experience, but are instead gained by guarding or parrying an enemy attack, and hitting an enemy with a deflected ranged attack by parrying (such as Wakka's blitzball). Less common to obtain Tech Points is by hitting certain large enemies, such as Darkside's head (instead of its hands), and Rock Titan in general. Lastly some enemies weak to a specific element also grant a Tech Point if hit with it, such as using Fire on a Blue Rhapsody.
Level up speed
Sora is the sole character with a variable level speed, which is dependent on the answers he gives to Tidus, Wakka, and Selphie at the start of the game in Dive to the Heart. Each answer itself is irrelevant, but the number of the answers chosen affects your Experience gain. Experience gain itself is limited to three speeds; Dawn, Midday, and Night (often called Dusk), per the message received after choosing your answers. Picking two of the first answers, or two of the last answers means you lock yourself in with either Dawn or Night, respectively. Any other combination leads to Midday, which is the most balanced leveling speed. Dawn starts out quite fast, making the early game easier, but slows down after level 40. Night on the other hand starts out quite slow, but speeds up after level 40. Notably Dawn also needs about 100,000 more EXP than Night to reach level 100 in total, but if you do not intend to level up much past level 40 it is still recommended.
Although reaching the maximum level (100) in Kingdom Hearts is entirely unneeded, as you can beat all optional bosses at around level 60 to 70, you might still want to do so for the abilities you gain at those levels. The first step to reaching level 100 will be to take the Night path at the start of the game, otherwise it will just take longer. Other than this you need to acquire any three of the EXP bonus items, ideally EXP Necklace, EXP Bracelet, and EXP Ring (or EXP Earring) through completing Muddy Path, synthesizing them, and defeating Unknown.
At this point you can choose to either repeatedly defeat the hardest repeatable optional bosses, these being Sephiroth, Rock Titan, and Ice Titan (whose projectiles award a lot of Tech Points if deflected while being equipped with all Tech Boost abilities). Other options include using Encounter Plus to repeatedly grind through areas with a large number of high level enemies (after rescuing Kairi), such as Traverse Town and relying on a high-powered Thundaga to quickly clear the field.
Another option that requires more skill is to juggle Rare Truffles 100 times for a large EXP bonus.
The way MP, Strength, and Defense work is fairly straightforward in Kingdom Hearts. What is less well-known is that Str and Def have 'softcaps', meaning that at a certain point they will no longer have a quantitative effect, making higher numbers moot entirely. Both stats cap at effectively 55, with the exception for Str against the optional bosses Sephiroth and Unknown, in which case it is 61. Any higher number means you do not deal extra damage. MP is similar, and caps at around 15 in terms of damage, but higher numbers still aid in Summon, Aero, and Stop duration, and the number of spells you can cast. AP also has an effective softcap at 92, as at that point you can equip all abilities you can acquire.