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 are reunited in the town of Altair, though Leon is nowhere to be found. 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 rebels' 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.
On your way out, you can find a Potion in a chest on the west side of the hideout. You can use Potions outside of combat from your inventory, but you must equip items to your characters in order for them to use them during combat. Each character gets two item slots, which can also be used to hold alternate weapons & shields. Note that Potions (and Hi-Potions) are twice as effective when used during combat.
Preparing for Adventure
FFII has unique systems for its time, and they may seem even stranger compared to more modern RPGs. The system is plagued by bugs and unintuitive interactions as well as important factors that are never mentioned (such as the magic penalty). I highly recommend studying the Party Planning and Game Systems guides before starting your quest in earnest. If you want to get the most out of the game, you need to play by its (obscure and often silly) rules.
The three characters currently in your party will remain there for the entire game, while your fourth character changes several times. Therefore it is wise to cover all your bases with the three characters you will always have. The system is flexible in theory but very restrictive in practice. If you don't have a specific strategy in mind, the simplest path is to make Maria your black mage (due to her high intelligence), while having either Firion or Guy play the role of white mage. The remaining character would act as 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: none of them starts with any skill in any weapons or shields. Don't feel obligated to stick with their starting equipment choices.
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 gil and weapons are expensive, but you already own a Broadsword, Axe, and Bow. None are appropriate for mages in the long term, but with only 5 MP each, no character is going to be casting a lot of spells in this early chapter. Any of the three black magic spell will cost all 400 gil you have right now. You should make sure all of your potential spellcasters have at least one spell before the next part of the game so they can start increasing their MP and relevant stats, so it may be prudent to save your money until you return from Fynn. Cure is a worthwhile investment of 200 gil because it will help you survive your early trials. If you want to swap your starting weapons for something else (perhaps more mage-friendly weapons), consider selling them to fund those purchases.
Armor is cheaper than weapons. An extra Buckler is a good idea for anyone fighting on the front lines. Leather Caps and Leather Gloves are cheap upgrades for anyone in the front row as well. Back-row characters don't benefit from defense or evasion, and thus gain nothing from any armor on offer here—even Clothes.
Items are all very expensive, though you might want to grab an Antidote or two when you have some gil to spare. They will be useful until you gain access to the Esuna spell and bring it to level 2, since your characters will not advance in any statistics while suffering from a permanent status ailment (including Poison drop than using them. (You will soon have access to Minwu's Esuna 3, which will last for a few game events, so none of this is needed urgently.)
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. Despite being in the back row during the first battle, Maria is now in the front row. If you want to have her use her Bow, move her back to the rear. 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.