Chapter 1 missable achievement - when you first start playing as the elf princess, you will have a few small objectives to complete before being able to fish:
*** Spoiler - click to reveal ***you must leave the room by first feeding the bird sunflower seeds, grabbing the mirror, putting it behind the music box, and opening the music box. this will trap the bird, then you can go to the library downstairs and once the medical book tells you to make a potion, you can start fishing for the Red Herring.
once you have the quest to get the fish, talk to Arbor the gardener to borrow his Fishing Rod, then go fish in the pond by the waterfall. your skill starts at 0, and goes up by 1 each time you fish. after a few points you can read the fishing book in the library to get +10 skill, then talk to Arbor about fishing for another +10. i was at around 25 skill when i got the fish, then just fished a few more times and got the Hat when i had about 30 skill.
some solutions are saying you need to have the pot of gold before you can hook the hat, but i did not have it. i did make the rainbow appear however, but did not dig up the gold: