Carwen is a brief stop between the Ship Graveyard and North Mountain. Not much happens story-wise, but the shops have useful upgrades.
Town Services
The inn costs 40 Gil. Use it. You probably need healing after the Ship Graveyard.
Check the armor shop for Iron Armor - solid defensive upgrade for your Knights. The weapon shop has Flails if you missed the one in the graveyard, plus basic elemental rods for your Black Mage.
The magic shop sells level 2 spells: Poison, Sleep, Silence for Black Mages, and Cura, Raise, Esuna for White Mages. Cura is essential; buy it now. Raise is useful for emergencies.
Town Exploration
Talk to the locals for hints about North Mountain. One NPC mentions that Lenna's dragon Hiryu was last seen heading there. Another warns about poisonous flowers on the mountain.
There's a pub with more NPC chatter. Nothing required here, but the worldbuilding is nice if you're into it.
Blue Magic Opportunity
If you have a Blue Mage, head to the fields outside Carwen. Tatou enemies can teach you Pond's Chorus, which turns enemies into toads. It's situationally useful and worth grabbing while you're here.
Next Steps
Head north from Carwen to reach North Mountain. The path is straightforward on the world map - just follow the road north until you reach the mountain entrance. Stock up on Antidotes before you go; the poison flowers on the mountain are no joke.