Below is the complete list of all 51 trophies for Final Fantasy XV. I will elaborate further on the requirements as soon as I unlock these trophies myself.
Trophy Name: The World Wanderer
Trophy Type: Platinum
Description: Collected all trophies.
Trophy Name: Insomnia's Waking Nightmare
Trophy Type: Bronze
Description: Completed the Prologue.
Trophy Name: Departure
Trophy Type: Bronze
Description: Completed Chapter 1.
Trophy Name: No Turning Back
Trophy Type: Bronze
Description: Completed Chapter 2.
Trophy Name: The Open World
Trophy Type: Bronze
Description: Completed Chapter 3.
Trophy Name: Living Legend
Trophy Type: Bronze
Description: Completed Chapter 4.
Trophy Name: Dark Clouds
Trophy Type: Bronze
Description: Completed Chapter 5.
Trophy Name: A Way Forward
Trophy Type: Bronze
Description: Completed Chapter 6.
Trophy Name: Party of Three
Trophy Type: Bronze
Description: Completed Chapter 7.
Trophy Name: Seaworthy
Trophy Type: Bronze
Description: Completed Chapter 8.
Trophy Name: Callings
Trophy Type: Bronze
Description: Completed Chapter 9.
Trophy Name: The Heart of a King
Trophy Type: Bronze
Description: Completed Chapter 10
Trophy Name: In the Dark
Trophy Type: Bronze
Description: Completed Chapter 11.
Trophy Name: End of Days
Trophy Type: Bronze
Description: Completed Chapter 12.
Trophy Name: Redemption
Trophy Type: Bronze
Description: Completed Chapter 13.
Trophy Name: Homecoming
Trophy Type: Silver
Description: Completed Chapter 14.
Trophy Name: Chosen King
Trophy Type: Gold
Description: Defeated Ifrit on Normal difficulty.
Trophy Name: Learner's Permit
Trophy Type: Bronze
Description: Drove the Regalia.
Trophy Name: Chocobo Jockey
Trophy Type: Bronze
Description: Rode a Chocobo. The chocobo rentals become available in Chapter 3 after you complete the sidequest "Friends of a Feather".
Trophy Name: Regalia Pilot
Trophy Type: Silver
Description: Flew the Regalia Type-F. First you complete the sidequest "Formouth Garrison" to get the Strange Engine. Then once you've finished the main story, take the Strange Engine to Cindy which completes the sidequest "Into Unknown Frontiers", and rewards you with the flying car.
Trophy Name: Quadruple Threat
Trophy Type: Bronze
Description: Equipped four weapon slots.
Trophy Name: Faithful Heir
Trophy Type: Silver
Description: Collected thirteen royal arms. The locations are listed below.
Trophy Name: New Power
Trophy Type: Bronze
Description: Learned first ability.
Trophy Name: Self-Improved
Trophy Type: Silver
Description: Activated 20 ability nodes. A tip on AP farming: The first thing you need is the Beast Whistle, an key item you get at the end of Chapter 3. Next you might want to wait until Noctis is strong to kill level 12 Voretooths in one hit. Now for the trick itself. Head to the field areas near the Disc of Cauthess and use the Beast Whistle to summon anywhere from 3 to 5 Voretooth. Use Warp Strike to defeat them and you should get 1 AP per kill. Repeat until you're satisfied.
Trophy Name: Self-Mastered
Trophy Type: Gold
Description: Activated 50 ability nodes. A tip on AP farming: The first thing you need is the Beast Whistle, an key item you get at the end of Chapter 3. Next you might want to wait until Noctis is strong to kill level 12 Voretooths in one hit. Now for the trick itself. Head to the field areas near the Disc of Cauthess and use the Beast Whistle to summon anywhere from 3 to 5 Voretooth. Use Warp Strike to defeat them and you should get 1 AP per kill. Repeat until you're satisfied.
Trophy Name: Angling Rookie
Trophy Type: Bronze
Description: Improved fishing level for the first time.
Trophy Name: Survival Rookie
Trophy Type: Bronze
Description: Improved survival level for the first time.
Trophy Name: Photo Rookie
Trophy Type: Bronze
Description: Improved photography level for the first time.
Trophy Name: Cooking Rookie
Trophy Type: Bronze
Description: Improved cooking level for the first time.
Trophy Name: Angling Expert
Trophy Type: Bronze
Description: Reached maximum fishing level. Keep in mind that the bigger the fish, the faster the skill levels up.
Trophy Name: Survival Expert
Trophy Type: Bronze
Description: Reached maximum survival level. You need to try and cover long distances on foot to get this skill to level up. A high survival skill level allows Gladiolus to pick up rarer items at the end of a battle.
Trophy Name: Photo Expert
Trophy Type: Bronze
Description: Reached maximum photography level.
Trophy Name: Cooking Expert
Trophy Type: Bronze
Description: Reached maximum cooking level. Ignis only cooks when you camp in the wild, and cooking a complex dish will improve this skill faster.
Trophy Name: Just Hangin' Around
Trophy Type: Bronze
Description: Performed first point-warp suspension.
Trophy Name: Brother-in-Arms
Trophy Type: Bronze
Description: Issued first ally command.
Trophy Name: Blind Spot
Trophy Type: Bronze
Description: Performed first blindside link.
Trophy Name: Noct You Like a Hurricane
Trophy Type: Bronze
Description: Initiated first link-strike after parrying an attack.
Trophy Name: The Power of Kings
Trophy Type: Bronze
Description: Called forth the Armiger for the first time.
Trophy Name: Magical Worker
Trophy Type: Bronze
Description: Crafted a spell for the first time.
Trophy Name: Black Mage
Trophy Type: Bronze
Description: Used magic for the first time.
Trophy Name: Divine Intervention
Trophy Type: Bronze
Description: Summoned one of the Six for the first time. Unlike in most Final Fantasy games, you cannot just pick a summon from a menu in FF15. Instead these things sort of have a mind of their own and only appear when certain conditions are met. Taking Titan and Ramuh as examples, Titan is more likely to come to your aid if all your other party members are KO'd. While Ramuh tends to show up the longer a battle drags on.
Trophy Name: High Five for Justice!
Trophy Type: Bronze
Description: Played JUSTICE MONSTERS FIVE for the first time. Be sure to look around when you visit diners in outposts throughout the game. Some of them have pinball machines that you can play for 10 gil. The trophy pops as soon as you quit playing.
Trophy Name: Immortal Photobomb
Trophy Type: Bronze
Description: Caught an image of Gentiana in a photo. Gentiana can show up in photos at any time from Chapter 5 onwards. You don't really have much control over this so just hope for the best.
Trophy Name: Spinning a Yarn I
Trophy Type: Bronze
Description: Completed first sidequest.
Trophy Name: Spinning a Yarn II
Trophy Type: Bronze
Description: Completed 5 sidequests.
Trophy Name: Spinning a Yarn III
Trophy Type: Bronze
Description: Completed 10 sidequests.
Trophy Name: Spinning a Yarn IV
Trophy Type: Bronze
Description: Completed 20 sidequests.
Trophy Name: Spinning a Yarn V
Trophy Type: Bronze
Description: Completed 40 sidequests.
Trophy Name: Weaving a Tapestry
Trophy Type: Silver
Description: Completed 80 sidequests.
Trophy Name: My First Hunt
Trophy Type: Bronze
Description: Completed first hunt.
Trophy Name: Tortoise Toppler
Trophy Type: Gold
Description: Defeated the adamantoise. The adamantoise hunt (Lonely Rumblings in Longwythe) becomes available in Hammerhead's diner after completing the sidequest "Let Sleeping Mountains Lie". As for the battle, the amount of time it takes to defeat the creature entirely depends on how strong you are. Lowering the difficulty from Normal to Easy also helps speed things up since the turtle won't be as strong. Not sure if difficulty affects the HP though, Also a word of warning on warp strikes. Noctis has a tendency to get stuck inside the adamantoise after warp striking. And while I'm usually able to get out by warping a second time, there were a couple of instances where Noctis got completely stuck, forcing me to reload my save. Also if you warp, make sure not to accidentally leave the battle area as it aborts the hunt, and will require you start over.