Moonglow Bay

You can unlock this achievement in chapter 2, during the mission "Some Friendly Advice". During this mission, you will get 3 new rods. With 4 different rods, 3 lures, and 2 baits (high-grade bait is available later in the game) you have more than enough combinations to unlock this achievement.

Even later in the game, you will be rewarded with the multi-lure as a reward for reaching the milestone of 20,000 shells spent on renovating the town. I don't know if this lure adds to the number of possible combinations, but you will most likely have unlocked this achievement well before reaching this milestone anyway.

Equipment
Rods: balanced, strong, poise, agile
Lure: jig, spinner, plug, (multi)
Bait: no bait, low-grade, high-grade

Now simply go to a fishing spot, and use 15 different combinations of rod, lure, and bait (e.g. balanced rod, jig lure, and no bait) to catch a fish. You can press at the fishing spot to swap equipment. You can also do this on your ship. In that case, you can swap equipment inside the ship.