Its central location means you'll spend a lot of time using Salamand as your home base, whether you're headed for the Semitt Falls or the Snow Cavern.
Salamand
Hilda asked you to find Josef, and he is waiting in the northeastern building in Salamand. If you ask him about [Mythril], he'll tell you that he has some leads, but won't discuss it until you prove yourself by rescuing the townspeople that are being held in the Semitt Falls. That's your next destination, but before moving on, take some time to explore the shops of Salamand.
The spells available in Salamand are an eye-popping 1,500 gil apiece, but three of the four of them are vital in the long run. You can find a Teleport Tome before Minwu leaves the party (and takes his own Teleport spell with him), but you should definitely invest in a Life Tome or two when you have the gil. Warp can also be very useful, though it is redundant with Teleport in many cases. The final spell for sale, Sap, reduces monsters' MP to ½ or less. This has some uses, but it is unfortunately bugged and won't work correctly against spellcasting monsters in the later game.
On the weapon front, the Longsword, Battle Axe, and Longbow all represent upgraded versions of your starting weapons. All three can be found in dungeons later in the game, but the damage upgrades they represent will be useful in the short term—if you can afford them.
The only new armor in town is the Bronze Armor, and the other bronze gear from Paloom and Poft can be purchased here as well. The Bronze Armor is strictly worse than the Copper Cuirass you could find in those towns, unfortunately. Unlike the weapons, you won't find any of this armor as treasure any time soon.
Now that you are seeing heavier armor, it is important to consider the evasion % penalties it comes with. Each character's agility (and thus base evasion %) increases based on their current evasion %, including bonuses from weapons & shields and penalties from armor. Your evasion % may not be high enough to actually avoid many attacks yet, but in order for it to get to that point, you should avoid letting its value get too low. You need at least 4 evasion % to have any chance of of gaining agility after battle. Characters in heavy armor should rely on a high shield level and the best available shield to offset the evasion % penalties.
To the Semitt Falls
The Semitt Falls are relatively close to Salamand as the crow flies, just across the mountains to the southwest, but reaching them involves a fairly long trip. There are two potential routes. The easier of the two is to head west until you reach a river flowing into the mountains, then follow that river back east to the Falls. This route has the extra advantage of being half traveled on Canoe, where you will not be attacked. You'll want to conserve resources on your way to the dungeon, so the fewer battles you have, the better.
The other route is long and almost entirely on foot. Head south from Salamand and follow the mountain ranges southwest and then west. You'll see a lake to the north, which is the same lake where the Semitt Falls entrance lies, but that entrance is at the top of the waterfall and you're stuck at the bottom. Instead, continue west, turn turn north until you reach a mountain pass that merges with the river that leads to the top of the Falls. There isn't much reason to take this route, but it's good to know about if you get lost while exploring the world.