Disclaimer: this answer was posted when Minecraft's newest version was 1.14., however, most of the information provided is relevant in later versions too.
There have been quite a few possible improvements to mining since most of the other answers were posted.
First of all, transportation from the top to the bottom: water elevators.
With update 1.13, water mechanics were improved, including a easy way of setting up super fast water elevators.
All you need on the way down is the old school method: water at the very bottom prevents any damage from a long fall.
On the way up, however, you can put Soul sand under a water elevator, which gets you to the top in a matter of seconds, and as a bonus prevents any drowning because Soul sand produces bubbles, as shown on the image:
Next, you can fit into a space that is one block high! There's two methods of getting this working.
- Using an Elytra to fly into the small space by flying into the one block. Here's the setup:
It may seem like you will have a hard time mining with your head in stone, but you do not take any suffocation damage and the point of view looks like this:
- Swimming into the hole. This method is very similar to the Elytra method. You place some water, start swimming in it and swim into the 1x1x1 gap. It is easier to set up than getting an Elytra, but it requires having to carry a bucket of water at all times.
Unfortunately, as of version 1.14, both these methods were massively downgraded, as there is a huge movement speed penalty (maximum speed is about the same as when sneaking), because the player remains in the flying/swimming position:
Even after the downgrade, the blocks revealed over time ratio is still slightly higher than digging a 2x1 corridor, and you don't have your head stuck in the block above anymore.
The final tip I have is using subtitles. This may seem like a dumb or useless feature to use while mining, but for me it works like a cave finder. If you are mining around Y coordinate 10, which most players do, you will be able to track nearby caves very easily. Whenever you see the subtitle "lava pops", you know you are near a cave. Look in the direction the subtitles tell the sound comes from, and start digging. The sounds that are coming from a distance and are detected by the subtitles are nearly undetectable by plain listening (at least for me).
So, if you like exploring caves, this is the way to massively boost your chances of finding one.