I have now thus far wandered the cold mountains of Skyrim for many hours, and I have seen many "ore" deposits that seem "mine-able". When I attempt to interact with these deposits I get a message that says something to the effect of "you need a pickaxe to mine that". I'll check the exact message text next time I'm playing. I've been to several towns, and I haven't seen a pickaxe for sale anywhere. Am I missing something? Where do I get what I need to mine ore?
Picks turn up commonly in and around mines - look for the crossed hammer icon on the maps (shown here):
You'll also see them in some of the shops in the villages.
In case you are fed up of searching for a pickaxe and really need one immediately (as sometimes happens with me when I'm searching for some particular item), you can just spawn one in your inventory using a cheat code. Although it's cheating I feel it is okay because after all it's just an ordinary pickaxe and you're saving yourself the grind of searching for it.
Here's what to do on the pc version:
Press the ` key (tilde key) to bring up the console
player.additem E3C16 1and enter.
Here E3C16 is the code for a pickaxe. Happy mining!