What is the most efficient way to find caves in Minecraft?

One tip to make cave searching much easier is to carry a piston and a form of power (button, pressure plate, lever, redstone torch etc.) around with you.

If you hear any monster noises, put the piston up against a wall (or floor or ceiling) facing into the wall (in the direction you think the monsters are). Power the piston, and if it actually works, you know there's a cave on the other side of that wall.

Keep in mind that this only works if the cave is 12 blocks or less away, any more and the piston won't work causing you to think that there's probably no cave.

While others give methods how to find single caves or explore to find bigger nexuses, let me answer directly, how to find BIG cave nexuses.

There are mods that let you peer through walls. This is a blatant cheat but it really lets you locate caves most easily. Another method to travel quickly, by boat or running. As new chunks are loaded, you can see through the floor before they do. That way you can spot places where a whole bunch of caves connect. It seems Minecraft uses some function of cavern density - while sometimes you find caves that run for chunks with very few branching, yesterday I encountered a couple of chunks that were literally such a mess of caverns, ravines and huge chambers, so that the remaining stone between them felt more like erratic walkways in a huge void, as there was considerably more air than stone.

As for frequency of cave appearance: look at the dashed cyan line of this graph: source

It's the frequency of air blocks per volume. In particular, caves between level 15 and 35 are nearly ten times as common as near surface (64). Although in this case the graph might be biased: I guess if you exclude "sea" biome the distribution would be more uniform.

Nevertheless, there's a rapid drop-off below 14, so don't seek them near bedrock.

I have no statistics, but, while digging, you can often hear faint monster noises deep underground. If you dig toward them, you should come out in a cave or dungeon.

I've also noticed that you can frequently find large caves that end up as ravines on the surface, and you can also find caves underneath ocean biomes.