Final Fantasy II has quite a different party setup—and leveling setup—from Final Fantasy. You will name your Characters at the beginning of the game (in the order you name them, their names are Firion, Leon, Maria, and Guy in the most recent translation), and three of them will be in your party at all times. The fourth, Leon, will eventually join the party and stick with it through the endgame, but you will have a number of other "fourth characters" (who you cannot name) join as the game progresses.
With the exception of Minwu, your extra characters are all better at weapon combat than spellcasting. Several of them can be easily converted into decent spellcasters, but it's usually best to leave them as fighters since it takes so much time to level spells. This means that your primary spellcaster(s) need to be among your three main party members. The standard setup is to give one character white magic, one character black magic, and leave the other as a fighter, but you don't have to follow this model. It is a viable option to give one character all the main spellcasting duties, though since any given character can only learn 16 of the 40 spells in the game, this requires some tough choices. Further, you're unlikely to have larger MP pools doing this, so in effect you're limiting your spellcasting. One spellcaster of each type is the simplest method to make work.
FF2 has a few system quirks that aren't mentioned anywhere which unfortunately prevent some of the more interesting party ideas you've no doubt imagined in a game with such an open-ended leveling system. The problem is that most weapons and armor have a harsh "magic penalty" which is applied to the accuracy of all spells. While some spells still have some effect even if they miss (especially damage spells), the long and the short of it is, you don't want your spellcasters to have any more magic penalty than they have to. This means no heavy armor, no shields, and few weapon options. There are a few tricks to minimize your magic penalty, but in the long run, any given character is either going to be a spellcaster or a fighter, but usually not both at once. It is worth noting that the magic penalty doesn't apply outside of combat, so having your main fighter learn useful out-of-combat spells like Cure, Life, Teleport, and Esuna is a good idea.
Note that spellcasters can get away with a high magic penalty in some circumstances. Damage spells are the least affected by the magic penalty, since they have low accuracy to begin with. Further, when using a spell that exploits a monster's elemental weakness, accuracy (and thus the magic penalty) is ignored in favor of automatic successes. However, both status spells and buffs are highly reliant on accuracy, so don't expect to get much out of them if you're casting with a high magic penalty. By the late game, your stats should be high enough that even damage spells have high enough accuracy that a magic penalty will significantly limit their damage (though by this point you should have powerful spellcaster equipment with a low magic penalty available).
So who should play what role? Because of FF2's flexible leveling system, where you get good at whatever you do most, you can realistically fit any character into any role. The only suggestion I would make is to make Maria your black magic user, since she has the highest intelligence score. Everyone in the party has the same spirit, so they're all good choices as a white mage. Guy is stronger and tougher than Firion, but less agile, making him a pretty obvious choice for your fighter. Truth be told, that may be the best choice, but I prefer to use Firion simply because he actually looks like he's wearing heavy armor, plus he starts with a shield. Having Firion with high evasion also leads to more ambushes. Be warned: Maria starts in the back row (though due to a bug she is back in the front by the time the game really starts) and has high intelligence, making her an obvious black mage. But she also starts with a bow, which is the worst possible weapon type (though not as bad as a weapon/shield combo) for spellcasters. So be careful!
Leveling
You may think that "leveling" doesn't apply to a game without levels, but you'd be mistaken. Final Fantasy II can be brutally difficult, and you'll need to keep your characters in tip-top shape to make it through. You can check the game systems page to get the details on how the leveling system works, but I will provide the short version here. Each of the two basic character types has its own considerations.
Fighter
The first thing you should know when playing the Famicom version of Final Fantasy II is that dual-wielding doesn't work as you might think. You will gain experience using your off-hand weapon, but you don't actually deal any extra damage with it. The second thing you should know is that shields are really good in this game. Thus, you definitely want your fighter to use a weapon and shield combination. Note that you will do a bit more damage using a single weapon without a shield, which can be useful when fighting heavily armored foes. It's a good idea to keep one of your fighter's item slots open, or carry alternate weapons for different situations. As for what type of weapon to use, that's up to you. I would avoid knives and staves, as they aren't generally powerful and you'll likely have your spellcasters using them. Bows are also out, since you want this character to be taking hits, not hiding out in the back row. Every other weapon type is perfectly viable, however, and it's a good idea to train with multiple weapons. In the long run, swords are best, but there are significant parts of the game where it will pay to have other weapon skills leveled.
As for armor, things are a bit more involved. A quick examination of the armor list reveals that body armor follows a very strange pattern. You'll find that certain pieces of armor, such as the Bronze Armor, are statistically similar to "cuirass" type armor (in this case, the Copper Cuirass). The differences are that the cuirasses, despite having the same defense, generally cost less, and have less of an evasion penalty. The obvious conclusion is that you should always favor cuirass armor, even for your fighter. The only hitch is that some believe wearing heavy armor causes your strength to rise faster. I have anecdotal evidence that supports this theory, but hard tests imply that it is not true. It is therefore probably a good idea to stick with cuirass pieces when buying armor, if only to save money. If you find heavy armor, or it's the only thing available, don't worry about using it, but don't go out of your way to get it, either. Fortunately, there is no such issue with head or arms armor, so simply use whatever gives the best defense. Keep in mind that multiple stat boosts to the same stat do not stack, nor do multiple resistances to the same element.
There are two basic styles of front-line fighter: the evasion fighter and the defense fighter. As a general rule, evasion is better—in addition to avoiding hits entirely (thus negating the need for defense), evasion determines turn order and, in Firion's case, how often the party is ambushed. Shields are key to high evasion, as is favoring lighter armor. However, especially at the end of the game, heavy armor isn't bad either. The Genji Armor has ridiculous evasion penalties, but huge defense bonuses to match. You might want to give it to Leon, since his evasion level starts out too low to make evasion rate very useful anyway. Finding a Dragon Shield from a Red Dragon or Blue Dragon can help with this setup, as it combines with the genji set to grant resistance to seven of the eight elements. Your evasion fighter should be able to maximize evasion with the Dragon Armor and Aegis Shield, a combination that grants all eight resistances. (The second-tier stat boosting items, such as a Gold Hairpin and Power Armlet, are good secondary pieces for maximizing evasion and overall offensive power.)
While any fighter is going to make a poor spellcaster simply by using a shield, there are a few spells that are worth using. As mentioned earlier, out-of-combat spells have their use, but some spells are even useful in combat. You will never get any spell successes, and your spirit and especially intelligence are likely to be low, but spells that exploit a monster's weakness will always be useful. Thus, the four basic elemental spells (Fire, Blizzard, Thunder, and Scourge) might be worth teaching to front-line fighters.
Spellcaster
The major consideration with any spellcaster is their armor. Very few armor pieces have a low enough magic penalty to be worth using. The main exceptions are the cuirass-type body armors, which replace the bracelets from Final Fantasy as standard spellcaster armor. Also as in FF1, you can give your spellcasters a basic Leather Cap and Leather Gloves, but they won't see better head or arm gear for quite some time. The defensive boost might be worth it if your spellcasters are in the front row. Late in the game, you'll find nice spellcaster-friendly armor like the Ribbon, Protect Ring, and White and Black Robes. These pieces don't have the best defense, but they offer resistances, magic defense, and bonuses to spirit and intelligence, all of which are much more important to a spellcaster in the back row.
Your spellcasters' weapon choices are considerably simpler. The only weapon types with acceptable magic penalties are knives and staves, though going unarmed is even better in this regard. Note that most mace-type staves have a 20% magic penalty, so you may want to avoid them, especially if your spellcaster is in the back row. I'm a big fan of the Werebuster in the mid-game, though. There are a few good reasons to give your spellcaster a staff over a knife. The Mage's Staff will cast Thunder when used as an item, which is a nice alternate attack, and the Wizard's Staff can also be useful if you like to take risks. Both are more powerful in the hands of a black mage than a white mage. The Healing Staff is deadly against Undead, and allows your spellcaster to heal without using MP, though it doesn't work from the back row.
Bows are another weapon option, especially for spellcasters in the back row, but they are tricky to use. They all have a massive magic penalty of 70, but only when actually held. It can be useful to keep a bow in reserve, switching to it only when attacking, and removing it when casting spells. By the late game, though, you should have enough MP to be casting constantly, especially once you have spells like Osmose and Swap that let you replenish your MP pool without using items.
The question of which row to put your spellcasters in is interesting. By the end of the game, they should most definitely be in the back row, since they have no need to attack. Having characters in the back row also helps your front-line characters gain evasion levels faster, which is important for protecting them from some nasty attacks late in the game. However, you may not want to put your spellcasters in the back right from the start. Early in the game, their physical attacks are still halfway decent, and you'll want them to get hit a few times to build up their HP from starting levels. Don't hesitate to move them to the back when you can get by without them, though.
Other Options
You aren't restricted to basic fighter and spellcaster options, though they are effective and easy to use. You can create a sort of hybrid character, if you're so inclined. However, a front-line caster is dangerous because they can use neither shield nor heavy armor, and will thus always be a liability, especially at the end of the game. Still, if you maximize defense while minimizing magic penalty, you can have a character that can both physically attack and cast effective spells. Because attacking can lower intelligence, it is easier to manage this with white magic. The weapon of choice for this character type is the staff, as they have good attack power with minimal magic penalties. The Excalibur and Masamune are also quite good, though you'll have to train up your sword skill to make use of them.
It is also possible to give one character both white and black magic. This can be problematic since your intelligence and spirit will not grow as quickly, making this character less proficient at both. However, it is a decent option if you want to have more physical attackers, or perhaps a dedicated bow user. The key is to choose spells that are always useful and not redundant. While you will generally have less MP than you'd like, once you get the Osmose spell, this character type becomes quite powerful.
While Final Fantasy II's system doesn't really allow for strange character builds, it does allow you to easily switch character roles. An evasion fighter can be brought up to speed as a front-line caster due to their high agility, or a black mage can be taught white magic late in the game. It's more effective to have one plan for the whole game and stick with it, but don't let that stop you from experimenting. You can make up for any deficiency, given a bit of time and effort.