To upgrade tools, make sure that they are ready to be upgraded by mousing over them. There should be an orange bar that appears. If it is at 100%, then the tool is ready to upgrade.
In order to upgrade an item, you will need 10 of the next level of material and have unlocked the Blacksmith. Keep in mind that your tool must be at full durability before it can be upgraded. You can repair your tool by talking to Danieru. The first upgrade will need 10 Copper Ore, the ones you get from the Mining Hill. Be careful: If you have a piece of stone and 100g, Danieru will likely take your item to repair instead, meaning you will not be able to upgrade until the next day. Go talk to him while holding the item you want to be upgraded. If the blacksmith is "open", click on chat option rather than the shop button. Your item will be taken from your inventory and it will be ready in 3 days.
To retrieve your item, talk to the blacksmith again, or chat if the shop is open. If the item is not complete, he will tell you something along the lines of "Your item will be done in x days".
!!WARNING!! When receiving items, they are sometimes sent to spots that are already in use. If this happens, the new item will be deleted when you try to do something with it, showing the older item instead. To be safe, ensure that the lower left spot is clear when retrieving your items.