I can't seem to find one. I was told there's one in Ember something or other, the cave to the west of where you start the game, but I can't find it anywhere. Can someone help me out? I have literally spent the last hour and a half looking.

There's a store where you upgrade, create weapons in "Riverwood"

"Edited" My bad.. I meant they're Pickaxes in Riverwood, The dude does smithing..

You can buy one off Alvor (the blacksmith) in Riverwood.

Go to Dawstar, in the mines

Dude there are free pickaxes in bleak falls barrow in some long cave strip with a river.

Ya I found one there first, I wasn't sure what it was for but I got it anyway cause I had a feeling I could use it

You could go to the top of the Throat of the world and on the highest peak there is a pickaxe with shock damage!