Where do I get keys?

Solution 1:

Mild spoilers for the story, notably what happens after defeating the first dragon boss.

You will need:

• the ability to walk on water (IIRC, I was around level 25 at that point)
• to be used to dodge attacks
• to have acquired and levelled up some skills, notably Cattrap¹ and Freezepaw
• magic-boosting gear, even if it lowers your other stats (mostly if you try it at lower levels, with a relativey low magic base stat).

When you're able to walk on water (after completing the Pawt City quests), go back to where it all began, namely your boat's wreck, south of Mewtown. You should be able to press E and trigger Spirry to sniff a trail. The trail leads south, to an island. You meet the developers there, who congratulate you for reaching the island. There's a quest board; take the quest. Sleep at the hotel to save your game.

The quest requires you to clear the island's cave. It's labeled as a level 1 cave, but that's just the developers messing with you; it's not so much a matter of level than it is a matter of having improved your skills.

The cave is full of mimics which can, and will, OHKO you. Hence the need to be able to dodge; you want to barge in, set a Cattrap, go back out of their range, set another Cattrap, slow them down with Freezepaw, land some sword hits to recharge mana, and dodge out before they land their attack. Pick magic-boosting gear, even if it lowers your health and melee stats (Arcane Robes, White Mage Robes, etc); the mimics one-shot you anyway, and your physical attacks only aim to recharge mana.

Repeat until all the mimics are dead; you will then find a key which can unlock all the golden chests. Note that some of them are hidden pretty deep inside the caves, and you may have to run into the cave map's "edges" to discover the secret paths.

¹_{Seriously, for this quest or any other, you really want to level up Cattrap as fast as possible. In my opinion, this is easily the handiest skill in the game, especially when you have several dragons and their minions to beat in a row, with no catnap available in between.}