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 in-world solution, you should look to one of the other methods of tracking distance which JavadocMD mentioned.
With current game version, hitting F3 will show your coordinates on the screen. You can then find how deep you are by comparing the bottom coordinates to the surface coordinates.
There's no easy way, as far as I know, you'll have to count.
I would suggest coming up with a system for reference. For example as you're digging down, you could place a distinctive block (one that wouldn't naturally be there, like cloth) every 10 vertical blocks traveled. Then it's easier to count later. Signs would also work, and then you wouldn't have to count 10s every time, just put the depth on the sign.
I'm sure some will object but there are also external mapper tools that can help with this kind of thing. In particular, I've found MCmap Live to be the simplest to use for this purpose because it has a slider that slices your map at any given level. If you don't use a mac, try one of these: http://www.minecraftwiki.net/wiki/Programs_and_Editors.
In practice, external mappers are much slower than just marking levels as others have suggested. However, in cases where I'm confused or lost, it's a tool I'm not above using. For instance, I recently found myself coming back from the Nether in a totally random underground location with almost no tools and I really needed to carefully plan my way up to the surface (to conserve my nearly spent diamond axe) or I was gonna end up punching through stone for hours.
The best to check your depth is to place torches, signs, wool, or iron, gold, diamond, or redstone blocks every 10 blocks or so. You can guess how far you are by the kind of blocks you've seen as well, but this is pretty iffy, as i've found iron and coal right before the bedrock. also, and this is only because you'd be mining, it's a good idea to have water with you or to have a 1*2 tunnel if you are tunnel mining.
I would put torches/signs every 5-10 blocks, and once you are at an adequate level for your desired ore, begin branch mining. but if you would rather, f3 is another good idea. the torch/signs method is a bit iffy because you might be doing it on an too high or low surface but i find that F3 can sometimes not work all that awesome.
Make a stack of stairs. Count out as many as you need if you want to reach a specific level (leave the rest on a separate stack) or just keep the full stack if you want to measure.
Build stairs down either until you run out of stairs (then you reached your desired level) or to the bottom (and your level is 64-[remaining stairs].
While this may take longer than just counting, the time spent on placing the stairs will be saved on running up them instead of jumping - two birds with one stone.