I started in Helgen, have walked to Riverwood and now Whiterun. I only have seen one mine/cave (other than in the Golden Claw quest) in my wanderings. Are there many? I had read there were and was surprised I couldn't find any on the way to Whiterun.

There are well over 100 dungeons, caves, mines, ruins etc..

don't worry, you will find plenty, just walk in any direction far enough and new locations will pop up on your map.

Thanks Frank