Tutorials/How to find caves
A cave is a place where you can find many resources, or it's just a little room. Sometimes you find them on the surface, others are buried underground. Some caves connect from the surface all the way to bedrock. These are the easy ways to find them (else than finding them by just digging rock). A tutorial on most of these can be found here.
 Surface Cave Entrance
When you go exploring the surface, you may find cave entrances. Many cave entrances can be found on the sides of mountains. Watch out though, mobs can be lurking at the entrance.
 Ravine Entrance
Ravines commonly intersect caves as well as abandoned mineshafts. Resources are a plenty, but beware of flowing lava. Don't worry too much about mobs unless you find a spawner or find an underground ravine.
 Sand and gravel
Gravel sometimes falls into caves, leaving air above. If you see gravel that looks like it has fallen, there may be a cave or mineshaft underneath. Also, sometimes, it is possible to find a cave when wandering through the desert. To do this, walk around until you find an odd 2 or 3 block drop in a small location that looks out of place. This is sand that has fallen. Dig stairs into it and you might find a cave. Sometimes there will be sand that fell down in the ocean, too.
Dungeons are places where monsters generate, and loot can be found within chests inside of them. Most dungeons are connected with caves. However, dungeons are quite rare and are not an easy way to find caves as most are found in a pre-existing cave and very few are found on the surface. When you find a dungeon, there will be a monster spawner. You can disable the spawner by placing torches on and around it.
(More detail on Dungeons can be found here)
Another way to find caves is to follow sounds from under the ground, or behind the walls in case you are already in a cave and trying to find another one. Practically, any sound you hear indicates an open area in the direction of the sound. This includes:
- Ambient sounds
- Lava noises
- Monsters' noises
- Monster Footsteps
- Flowing water
- Fire (comes from Abandoned Mineshafts' wooden structures, which occasionally catch fire from lava)
Determining the exact direction of the sound is tricky. Few hints:
- If the sound does repeat time to time, try turning your head (ingame) while listening. First make it sound in one of your ears (in headphones/speakers) louder than in other, that would give you general direction. Second, try finding the exact direction by turning your head (ingame) so that its volume is as even as possible in both of your ears (in headphones/speakers).
- It is usually unclear if the sound comes from below, or from above. Try digging two blocks down, and climbing two blocks up. Compare the volume. The closer you are to the source of the sound, the louder it gets. If it is still the same, then the source is probably on your original level.
- If the sound does not repeat, note that you can hear sounds generated from no further than 16 blocks away (for most of the sound sources). That means you can dig no further than 16 blocks away to find another cave.
Also, a sound like the howling wind can be heard when the player is in close proximity to a cave or other opening. If you seal up your basement you'll hear this sound as well. The sound is created by any cavern in which you are within 3 blocks of, and it is incredibly hard to pinpoint. The easiest way to find the caverns is to simply smash through walls.
This is a simple and effective technique. Simply place down pistons facing the wall(s) you are looking for caves behind and power it with a redstone torch. If there is an empty space within 12 blocks the piston will push. Simply dig in that direction and you should find a cave!
 X-ray machine
NOTE: This method does not work in the current snapshots and beyond, due to the player suffocating in TNT while not seeing any vision at all. Albeit because of some servers still use an outdated version, it will be kept on for legacy purposes.
Glowstone/TNT/Redstone Blocks/Leaves can be used to make an X-ray machine to see through solid ground and make caves appear as black. Simply set up a piston with a lever to push Glowstone/TNT/a Redstone Block/Leaf into your head and legs. You can also place a fence, and then place Glowstone/TNT/a Redstone Block/Leaf on it so it gets in your head and feet. There are various tricks to improve this. This is best done above ground away from ravines, cave entrances, and man made underground structures.You could also download an X-Ray texture or as of 1.6 resource packs. Note this is considered cheating by most of the community, and will get you banned from most servers. The easiest X-Ray machine to make can be created simply with some iron, sticks, sand and gunpowder, which should take no more than 10 minutes to gather. Easy to transport, here is a link to a tutorial on how to make one: