It says to click on a port to change it. On the ship configuration screen, I can't see any port buttons to click. All I see is my ship on the left side but there are no buttons to change ports.

Maybe it's a bug? Or maybe I am just stupid. I have stuck on the mining quest for a week now because I can't switch to the mining laser.

The "port" is the box shown over the icon of your ship. From the Ship -> Manage -> Configure Ship -> Addons tab, first select the port containing the item you wish to unequip, then click "Unequip"



Then, select the item from your station inventory you wish to equip, select the port you wish to equip it to, then click "Equip"