As with quests, there are simply too many fishing-related achievements in the game to provide useful details here. Instead, I recommend you take a look at this List of Fishing Achievements to see which ones are appropriate for your skill level and faction.
Nothing shows your dedication to a goal in WoW like a title. By completing the achievement Accomplished Angler you will be granted the title Salty. Everyone will then know you have invested countless hours catching all manner of fish (and junk!) to earn a title few dare to attempt.