You need to keep finishing villagers' requests. Some of them will give tier 2 dishes. I myself only unlocked three of tier 2 dishes. Lot more dishes to unlock.
Pom needs to pay first debt (2000 coins). Then the next day, Pom will receive the postcard from her parents telling her about Mala Caverns. And when you go to World Map, you'll see Mala Cavern is open.