You have no problems if you have just spent your night in a cave. If you haven't, the easiest way to do so is to roam the overworld a little and look out for exposed cave entrances. Sometimes they end immediately, but sometimes they develop into big cave systems.
If they seem to end right away sometimes there will be a continuation about 2-8 blocks farther back and below the cave's end; mining in these directions (8ish blocks back and down, squared [meaning 8 blocks in a row each layer dug]) will reveal this if it is the case. Digging away gravel or dirt can also expose cave extensions, but you may go through a few shovels that way.
If you can't find any all day, just sleep away the night, replenishing your resources like wood and food as you use them. If you hear suspicious, hostile noises, just try to listen and follow them.
