When you start a new game of Final Fantasy II, the first thing you'll do is name your characters. They don't have names already filled in, but in the order you are allowed to name them, they are canonically known as Firion, Leon, Maria, and Guy.
Once your Characters are named, you'll be thrust into a brutal battle that sees your party quickly and unceremoniously defeated. There's nothing you can do to affect the outcome. Three of the four party members, excepting Leon, are reunited in the town of Altair. Princess Hilda denies your initial plea to join the rebellion against the Empire that invaded your home. Make sure to speak with her again and use the Keyword system to learn the password [Wild Rose]. Ask Hilda about [Wild Rose] and she will give you your first task: go to Fynn to search for Leon. Before leaving, take some time to gather information and gear up your characters. You can also find a Potion in a chest on the west side of the rebel headquarters.
Preparing for Adventure
FFII is not like modern RPGs. Indeed, it isn't much like its contemporary RPGs, not even FFI, except graphically. The game is frankly plagued by bad systems and design decisions. As such, if you intend to play it, you will want to be well-informed. I highly recommend studying the Party Planning and Game Systems guides before starting in earnest. If you want to get the most out of the game, you need to play by its (obscure and silly) rules.
The three characters currently in your party will remain there for the entire game, while your fourth member rotates throughout the game. As such, you'll want to cover all your bases with the three characters you have. While theoretically the free-form leveling system would allow you to customize your party any way you like, in actual fact your are quite restricted in your options. If you are not planning a specific strategy, the simplest route is to make Maria your black mage, while having either Firion or Guy play the role of white mage while the other becomes a front-line fighter. However, at this point you lack the funding to do much beyond having all three characters fight.
One last thing to note when deciding on what to do with your Characters: you can ignore their starting equipment distribution entirely. All three of them start with zero skill in every Weapon and Shield. Only their starting stats matter at this point.
Altair
Altair is a typical town with an Inn, a Sanctuary, and all four types of shop. FFII has a few quirks compared to its predecessor. Every inn has the same price structure, which is based on how much HP and MP you need to recover. Sanctuaries can resurrect dead Characters for free. And every item Shop sells the same 12 Items across three counters.
You start with a mere 400 gold and Weapons are expensive, but you already own a Broadsword, Axe, and Bow. None are appropriate for mages, but with only 5 MP each, no one is going to be spending all their time casting just yet—especially since any black magic Spell will cost all 400 gil you have right now. Your white magic user should pick up Cure, though. A Knife or Staff can work as a mage weapon, though fighting unarmed is also a good option. Your front-line fighters will spend most of the game using swords, axes, and spears, so picking up a Javelin when you can afford it will allow you to start learning all three weapon types.
Armor is cheaper than Weapons. If you have two front-line warriors, buy an extra Buckler for the second one. Leather Caps and Leather Gloves are good upgrades for anyone liable to be attacked as well. Back row Characters don't benefit from Defense or Evasion, and thus gain nothing from any Armor on offer here—even Clothes. That said, you'll likely want to bring everyone to the front row early on to make sure they gain Maximum HP, so having Armor available is a good idea.
Items are all very expensive, though you might want to grab an Antidote or two when you have some more cash. Eye Drops are also good to have on hand, though you won't run into Darkness for some time to come. These Items will be useful until you gain access to the Esuna Spell, since your characters will not advance in any stat while suffering from a permanent Status Ailment. Due to your limited inventory and their inflated prices, you may end up selling more of the various remedies that Monsters drop than using them.
Once you're geared up, you should head north for the town of Gatrea. At this early stage of the game, you'll want to fight lots of battles to train up your stats and earn gil for gear upgrades. Also note that you can adjust your Characters' rows by pressing Select on the map screen. Contrary to the first battle, Maria is now in the front row just like everyone else, so make sure to check this. Use the nearby towns to your advantage, frequenting the inns whenever needed. Make sure to use Cure before resting at an inn, since it will save you money as well as help level the Spell!
Gatrea
Gatrea is a small town north of Altair. There is no Spell Shop located here, but the Weapon and Armor Shops have slightly different goods than those in Altair. You can pick up a second Axe or Bow here, as well as a third set of Leather Armor. Otherwise, Gatrea serves mainly as a waystation on the way to Fynn. A man in town will tell you about a bar at the edge of Fynn that you'll want to look into.
Once you have a few battles in the World Map Fynn Area under your belt, you're ready to head towards Fynn itself. None of the Monsters here are noteworthy, and all are likely to be defeated by one solid hit. Hornets inflict Venom with their attacks, a temporary form of Poison which you don't need to worry about. Gaining skill levels isn't necessary at this point, though you may want to spend some time getting used to the nuances of the battle system, such as monster rows and temporary status ailments, before you move on. If you have not played FFI, you'll also want to get used to the fact that if a Monster a Character attacks dies before the attack happens, it will not be retargeted at another Monster. Distributing your attacks is important to keep battles moving quickly.