For the armor achievement you will need to get the blacksmith to level 7 (level 8 for weapons). In order to level up the blacksmith you need to use grindstones which will advance an XP bar underneath him.
Whenever you enter caves if there are visible mining nodes you can attempt to mine them vs a die roll. You can also purchase grindstones in town for 25 gold each.
You can significantly reduce the cost to blacksmithing by 90% by purchasing Furniture Right: Sword and Shield in the shop accessed by pressing X along with 4 Greater Molten Rings from the item shop in town (your character needs to be level 16 to equip them). Before attempting to upgrade your character equip all 4 rings then proceed.