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:
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.