You can mine the following items:

Iron ore is used to craft iron ingots. These ingots are the ingredient to craft steel ingots . Silver and gold ores can be smelted into respectively silver and gold ingots.

Go into the mine with the Mining Pickaxe and hit the stone rocks. The ores are tricky to find and will be randomly spawned. Watch out for mine collapses. This happens frequently and is a reason to avoid this hard work.

Upon succesfully collecting ores you'll need to smelt these. Climb the stairs outside to reach the top of the furnace. Now put enough ores into the minecart to create an ingot. Now push the contents of the minecart into the furnace by reinteracting with the minecart. Now you'll see the minecart animated to throw everything into the furnace. In order to complete this progress, you'll need to keep your camera on the original position of the minecart. Use ~-key to rotate your camera. This applies to all machinery.

A minigame is the last barrier before you can succesfully smelt ores into ingots A fast moving arrows goes forth and back. Click 'Finish' if the button reaches the section between the two markers to succeed into smelting. 

Smelting was historically a very tough and difficult process. The result could easily be broken when the furnace didn't had the right temperature. The Industrial Era has brought us techniques and furnaces which can reach immense temperatures. But for now you'll have to work with medieval techiques!