From what I've read, fishing is raised by two things: by the amount of fish you wriggle out of the water, and the rarity. The rarer the fish, the more experience, but you can also weigh that by how many fish you can pull out too.
Go to the small store on the southern tip of the Vesperpool (around chapter 7) and buy the best rod/reel you can afford (or get the Booster Pack + with the Season Pass, as the two angling items provided are only a little bit better). While neither of these are the absolute best, they're decent enough for what you need them for. Super Baleen or Dragon Beards would be preferable (especially DB, since it has 2k durability) for the line, and as for a lure, just use whatever is abundant around you (either fishing near a store, or use whatever is beneficial for the fish near you).
All you have to do now is just fish, fish, and fish. You'll be casting so often you might want to mute the game or characters at least for how often they remark and comment on things (especially if your line durability goes below half). Some reports say that two straight hours of fishing will end up getting you max level.