@ Oddey, very nice
@ Enchilado, I can't possibly back up what you say. If you use my method of digging down diagonally, you're guaranteed to live. In your method, you only see the four squares around you. In my method not only do you see more surface area, you have a safe way back.
I do agree with your idea of carrying dirt. Personally I prefer sand, since it sticks out like a sore thumb in a cave. You'll probably need between 15-30 blocks of sand. Try to make a stairway out of stone if possible.
If you find lava, I recommend putting it out with a water bucket if you have one. I personally always recommend carrying water with you. You should keep a backup tool in case you run out of all of them. When you're starting out, you can bring a workbench to your cave and then just construct lots of stone tools again to dig further down.