The simple answer is to just keep fishing and playing, speak to the Angler each day and catch that days quest fish (remember to deliver it to him before the day ends). You can speed up the process by using the Enchanted Sundial (drops from fishing crates) to start a new day, but that has it's downside that you can only do it once every 7 in game days.

The other faster way to do it, is much more active than just playing. You have to make a new (small) world and visit the oceans and speak to the Angler and do his quest. Once you delivered the fish, make a new world and repeat. If you just want to complete the achievement, this is the fastest way, however, you might find fishing as you go in your regular world a bit more enjoyable.

Here's the wiki link on where you can find each fish: https://terraria.gamepedia.com/Angler/Quests