Forming a Party
The first thing you will do when starting a new game is form a party. There are six character classes, but you only have four party members, which means you can't use every class in a given playthrough. You are free to use the same class multiple times, so if you want to play with four White Mages, feel free. I am not big on the "party challenge" aspect of FF1, so I will just be giving advice based on a normal playthrough.
The Standard Setup
Most often you will want a party following that old D&D tradition of tank, damage dealer, healer, nuker. There are quite a few setups that follow this general mold. The most important role to fill in any party is that of the tank. Your top character will have 50% of attacks directed at him, he needs to have a lot of Defense. The only class that qualifies as a tank throughout the whole game is the Warrior. They can use all the best armor, and dish out quite a bit of damage to boot. You can play without a Warrior, but doing so will make your life very difficult at some points.
The damage dealer role is filled by a much wider variety of classes. In the default party, it will be the Thief. Thieves are decent, but honestly are outclassed (no pun intended) by every other choice. As Thieves they have light armor, mediocre weaponry, and no particular statistical advantage except being very good at running away. Running away can be tactically sound, but it's not the most impressive skill. However, when upgraded to a Ninja, this class really shines. They can suddenly use most armor and weapons, as well as level 1–4 black magic which includes fantastic support spells like Steel and Haste. Even so, their gear is still second to the Warrior, and a real caster job can use the same spells. The Ninja is fun, but not vital.
The second choice for a damage dealer would therefore be another Warrior. They are expensive to maintain, though the second doesn't need to be quite as heavily armored as the first, but they dish out a ton of damage. There's only one Excalibur, but even if you give someone else the Masamune, a second Warrior is a force to be reckoned with. If you want a bit more finesse than brute force, a Red Mage can work as your damage dealer, most likely prompting a very magic-heavy party. Red Mages shine early in the game, putting Thieves to shame, but later in the game their ability to dish out damage in combat begins to suffer greatly, so this is really only a magic-lover's option.
The most traditional damage dealer, however, is the Monk. They're kind of pathetic at the beginning of the game, but by level 7 or so they start to come into their own. Their armor starts out fairly weak and doesn't get much better over time, but they get a ridiculous number of HP to compensate. And they dish out extreme amounts of damage unarmed. With double the hits of any other class and a higher crit rate than any weapon, Monks are truly the premier damage dealers. Honestly, the best reason not to use one is that they're not all that interesting. They punch things really hard with no real effort from your end. Effective, but kind of boring.
The second most important role to fill is that of the healer, and there are really only two choices here: the White Mage or the Red Mage. Obviously the White Mage is the best overall healer, with access to the Heal spells as well as the Cure spells, as well as being the only class that can cast Life2. But if you want to focus squarely on offense, the Red Mage is a passable healer that is much better at hurting things (other than Undead) than the White Mage. But unless you plan to use two Red Mages or go without a nuker, you're probably better off replacing your Black Mage instead.
Which brings us to the final role, the nuker. Because of its old school D&D "spells per day" system, there is no job in Final Fantasy that can just cast spells all day long. Later in the game you have items that can do this, though they only go up to tier-2 spells so your nukers are not made completely obsolete. But unlike later games in the system, nukers are really not about dishing out large damage in FF1. Rather, they're about dishing out damage to a large number of targets. When you're fighting a group of 9 monsters, it helps a lot to be able to take them out in one spell. You have two choices in this area: the Black Mage or the Red Mage. In terms of spell availability, the Black Mage only gets one good damage spell the Red Mage doesn't, but that spell is Flare. Flare is underrated by some, since the late-game enemies you'd really like to hit with it generally have very high Magic Defense, but used correctly it is the ultimate crowd-clearing spell. Still, you can live without it, and most of the Black Mage's other exclusive spells are instant-death spells (which are never fan favorites), so many suggest using a Red Mage as your nuker. After all, they can dish out decent damage even when not using one of their high-level spell slots, and can even cure a bit in a pinch. Personally I still prefer the Black Mage since at the end of the game either class will mostly be using item-based spells anyway so the Red Mage's attack advantage is minimized.
Non-Standard Setups
Of course, you don't have to follow the standard setup. I would never advise going without a tank or a healer, but you can give it a shot. Some wacky parties can work very well, such as four Warriors, who rarely take damage that needs to be healed anyway (at least from normal attacks). The old Nintendo Power guide suggested a party of Warrior, Monk, Thief, and Red Mage, which is about as melee-heavy as you can go. You might have some fun with a Warrior and three Red Mages, as well (a very good party in the early game). One party that may seem powerful but should never be attempted is a party of four Monks. At max level, they are unstoppable, but they have serious problems getting through the early game because they just can't hurt enemies with high Defense. Feel free to experiment, but keep in mind that the crazier your party lineup is, the more problems you're liable to run into.