Is there a way to see how deep I am?

Is there a way to see how deep a hole I've dug in Minecraft, besides counting the number of squares I've dug through? I realize at some point there is bedrock so if I can't see how far down I am, can I see how far from the bedrock I am?

You can view your distance from the bedrock layer by pressing F3, which will show your current coordinates on screen. This isn't quite the same as knowing how far you are from the surface, but it's a good way to get a quick estimate of how deep you are.
Before the F3 feature was added, the best way to estimate your vertical position in the world was to pay attention to what kind of ore you find as you dig. (This is still very useful knowledge to have, for obvious reasons.)
If you prefer not use F3 to get your exact coordinates and want a purely inworld solution, you should look to one of the other methods of tracking distance which JavadocMD mentioned.