How do I get the lantern for the cave?
Perhaps the light rod from the alchemical vendor? once you have access to the beach, you can find an empty lantern, atleast thats where i found one. its in a golden chest, so pray to RNGesus for good luck...
then you need to speak to the bartender in the tavern which points you to the mages guild, the other guild in town, where they have a fusing machine. put both the light rod and the empty lantern inside and you have a light source.