Moonfish are one of the primary side quests that was added into Skies of Arcadia: Legends. They are much similar to Chams in the sense that you find Moonfish in cities and dungeons and you're alerted to their presence by a loud beep. The closer you get to the Moonfish's location the louder the beep becomes until it eventually turns into a shrill at which point you know you're standing in the right location to catch the Moonfish.
In order to catch a Moonfish you'll need to zoom into first person view with Y and search around the sky for it. Once you've found it your target will lock onto it and you'll hit A to bring it to you. That's all there is to it! If you're not standing in the right location it can be hard to even see the Moonfish - only one part of it will be visible and that's the glowing ball atop its head.
There are a total of 24 Moonfish in the game and collecting them will net you various different rewards when you turn them into the person that asks you to collect them... Which is Doc - a medic that's found in a boat around Sailor's Island which is the second island you come across in the game. He's found in a small ship with red and white flags. If you're having trouble locating him I show his boat in the screen shot below.
Just fly up to his ship and press A and it'll trigger a cutscene with him, a little girl with blonde hair and the weird creature that is eating the Moonfish. Each time you feed this creature some Moonfish he'll hack up some items for you which Doc lets you keep as a reward.
Normally you get a variety of armor and weapons which are good for your level (assuming you collect every Moonfish as it becomes available). The final four rewards that you get for collecting the Moonfish are some of the best ones. You can view all the rewards for this quest over on Gamefaqs here... But here's the best ones:
Moonfish 23: Hydra Wing (Aika's best weapon)
Moonfish 23: Silver Arm (Drachma's best weapon)
Moonfish 23: Serpent Strike (Enrique's best weapon)
Moonfish 24: Moon Hamachou Discovery Location which is worth 40k Gold.
Putting all of the Moonfish locations on one page made my internet browser load slowly because it contained too many pictures. Due to this, I have broken the Moonfish Locations up into three separate guides each of which provide you with much detail about how to locate the Moonfish as well as a picture of exactly where to stand.
On this page I go over the basics of each Moonfish's location, such as the area they're located in and a minor amount of detail about how to locate them. If that's not enough information for you I strongly recommend you follow my link for each location to the actual guide that I have laid out.
Moonfish Locations 1 - 9
Moonfish 1: Sailor's Island near Polly's Tavern.
Moonfish 2: Sailor's Island second floor of the Inn.
Moonfish 3: Pirate Isle on the Lookout at the top of the town.
Moonfish 4: Pirate Isle in the town's port near your ship.
Moonfish 5: Maramba northern end of the docks.
Moonfish 6: Maramba ride a Dhabu to a nearby building.
Moonfish 7: Temple of Pyrynn just off the boulder maze.
Moonfish 8: Temple of Pyrynn found after dumping water onto the lava.
Moonfish 9: Horteka found behind the elder's house near a cliff edge.
Moonfish Locations 10 - 18
Moonfish 10: Horteka outside the tavern.
Moonfish 11: Moonstone Mountain shortly after meeting Centime.
Moonfish 12: Lost City of Rixis area with moving platforms.
Moonfish 13: Nasrad found on the docks.
Moonfish 14: Daccat's Island Pillar Maze on Vyse's side.
Moonfish 15: Crescent Isle the island's port.
Moonfish 16: Crescent Isle the balcony adjacent to the meeting room.
Moonfish 17: Esperanza the top of the lighthouse.
Moonfish 18: Esperanza north by item merchant.
Moonfish Locations 19 - 24
Moonfish 19: Yafutoma near the King's hut.
Moonfish 20: Mount Kazai towards the end of the dungeon.
Moonfish 21: Tenkou Island at the top where you meet Diago.
Moonfish 22: Glacia enter from the Lower Sky entrance.
Moonfish 23: Dangral Island when you first arrive - right nearby.
Moonfish 24: Dangral Island in the port near the helm of De Loco's ship.
The Doc/Moonfish side quest is intertwined with the Piastol side quest in Skies of Arcadia: Legends. If I explained how or why it would contain a bunch of different spoilers so instead all I am going to say is one thing. If you ever arrive at Doc's Ship at some point in the game and Maria tells you that Doc is currently on a house call - it's likely because you still need to complete the Piastol "Angel of Death" event.
You can find Piastol's ship in the same area about as Docs - just a little but more north. As far as I know only Piastol's third event will stop you from interacting with Doc on his ship. Usually this will be about 3/4 of the way through the game, around the same time you complete this Moonfish side quest.