There are three ways to obtain melon slices: 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.
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.
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.
gaming.stackexchange.com/users/17801/w1nger1
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.
gaming.stackexchange.com/users/18/Kevin Yap
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.
gaming.stackexchange.com/users/None/
Melons could be challenging to find, depending on your world's seed and your luck.
If you do find one, you can start a melon farm by planting seeds and harvest your own melon.
Usually, people use melon mainly for brewing purposes.
It is rarely that a player would eat a melon slice to fill its hunger.
This is because the hunger bar has another "hidden" bar, so even if you are only "slightly hungry" and eat a big steak, the steak will not go waste.
Here's why:
There are four fields in level.dat which are related to hunger: foodLevel ranges from 0 to 20 and is represented by the player's Food Bar.
foodSaturationLevel is an invisible additional hunger variable that is depleted before main foodLevel value.
Eating any food will also add some to this variable.
Note that this cannot exceed foodLevel.
The Food Bar jitters when this equals 0.
foodTickTimer increases with every tick when foodLevel is either greater than 17 or equals zero.
When foodTickTimer reaches 80, it resets to zero and then heals or deals one point of damage to the Health Bar, respectively.
foodExhaustionLevel ranges from 0.0 to 4.0 and increases with every action the player takes.
When the exhaustion level reaches above 4.0, it will be subtracted by 4.0 and subtracts 1 point either from foodSaturationLevel or, if foodSaturationLevel equals zero, from foodLevel. As you can see, if your foodLevel just went down from full (20) to "slightly hungry (19)"
, it implies that your foodSaturationLevel has went down to 0.
Now suppose you eat a slice of melon, you'll restore food points plus a mere 1.2 foodSaturationLevel .
When you eat a steak, however, you'll restore 12.8 foodSaturationLevel , which means you will get hungry again much later.
gaming.stackexchange.com/users/35931/user1032613
I think you can find melons in jungles and of course mine shafts
gaming.stackexchange.com/users/58884/user58884
You can only find melon seeds in abandoned mine shaft chests and dungeon chests.
gaming.stackexchange.com/users/23428/unknown
I'm frequently finding myself in situations where I am barely hungry with 9\10 Drumsticks filled, but also with low health. I don't want to waste cooked meat on filling that last Drumstick to start my health regen.  Melon slices, which fill 1 slot of the hunger bar (or one Drumstick), would be perfect for keeping me full.  What's the fastest way to find melons?
What's the fastest way to find melons?
gaming.stackexchange.com
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gaming.stackexchange.com/questions/42133
gaming.stackexchange.com/users/5029/John the Green
You can find melon seeds in an abandoned mineshaft, trading with a villager or crafting a melon you already have. You can then grow and harvest a melon block.  Finding and producing melons can be done mainly by stumbling uppon a mineshaft, where the player will find melons or melon seeds.
The community suggests few ways of finding or producing melons.