What's the fastest way to find melons?

There are three ways to obtain melon slices:

1. Grow and harvest a melon block.

   • Locate an abandoned mineshaft, they commonly intersect with a ravine, so ravines are a good place to look for them.

   • Look for chest within an abandoned mineshaft. According to the wiki, the chests there have 13% chance to contain 2-4 melon seeds.

   • Grow melon on farmland using melon seeds.

   • Harvesting the fully grown melon block yields 3-7 Melon Slices.

2. Trade with farmer villager.

   • Find an NPC village. They can only be found in desert or plains biome.

   • Obtain emeralds by trading with villagers or from mining (Emerald Ore is very, very rare).

   • Look for the 1 emerald for 5-8 melon slices deals offered by farmer villagers. They are the ones dressed in brown robes.

   • If no villagers offer that deal, you can grow more villagers by building more houses (doors). Alternatively, you can kill villagers to make room for new villagers to spawn.

3. Find a jungle biome. If you explore a jungle biome enough you could find melon blocks. Break these and get 5-8 melon slices per block. From there you could craft the slices into seeds and start a farm.

It depends on your situation to determine which way is faster. Finding and exploring abandoned mineshafts will be more fun but does not guarantee you will find melon seeds. Discovering an NPC village guarantees that you will eventually have a villager that will trade melon slices. As a surplus some villagers might have some really nice trades, but it gets really, really annoying when you have a lot of villagers.

Either way, to sustain a constant supply of melon slices, the best way is to have a melon farm. As mentioned above, melon seeds can be found in abandoned mineshaft chests. They can also be crafted, 1 melon seed from 1 melon slice.

Additional Note: The 13% chance to find 2-4 melon seeds in an abandoned mineshaft chest sounds very slim, but there is a high chance that there will be more than one chest in an abandoned mineshaft. They are big places after all.

Well, the only way to find Melons would be to stumble upon an Abandoned Mineshaft, and hope some of the chests have Melon Seeds. From there, Melons are extremely easy to farm: like Pumpkins, when you plant the seeds, the stalks that grow will not disappear after harvesting the melon they grow, so assuming you make a large enough farm, you will have a fairly constant supply of melons.

As of version 1.4.2, there are quite a few ways to acquire melon seeds. As listed by the Minecraft Wiki, you can find Mellon seeds by either finding one in an abandoned mineshaft chest, trading with a villager, or 'crafting' a melon slice you already have.

Note that the Minecraft Wiki is outdated most of the time, but from my experiences, these are currently the only way to acquire melon seeds as of version 1.4.2.