This achievement is only possible after you have cleared the game for the first time and have the ability to use chapter select, as there are two paths in chapter 9 which makes doing all of them at once impossible. The first set of side quests you encounter are in chapter 3 and they are:
Chadley’s Report:
This is completed by using the Assess Materia on two different enemy types.
Lost Friends:
Completed by finding all 3 of Betty’s lost cats, the first one is found just outside of Seventh Heaven, the second is found on the path to the train station, and the third is found after going through a small gap after a woman in the town asks you to go and get something from a cat for her.
Rat Problem:
Completed by first killing the first wave of Wererats, after you kill the first wave, you will have to leave the area they are in for a while and come back at a later time to kill the Doomrats.
On The Prowl:
Before you can access this quest, you must do the Ray Problem Quest prior. Completed by killing the Shinra Dog that has escaped.
Nuisance In The Factory:
To complete this quest, you will have to venture inside the factory and kill the two groups of Lesser Drakes inside.
Just Flew In From The Graveyard:
Before you can access this quest you must complete the Nuisance In The Factory quest. Inside the factory you will find a locked door, this door requires that you find the key for it, the key should be in some boxes next to a vending machine. Once the door is open you have to kill the Cerulian Drake inside.
Chapter 8:
The Mysterious Moogle Merchant:
Completed by purchasing the Moogle Membership Card from the Merchant with Moogle Medals, the merchant can be found in the children’s hideout.
Weapons On A Rampage:
Completed by killing two groups of Mark II Monodrives in the scrapyard. If they appear to have a shield around them, it can be removed with magic abilities.
Paying Respects:
This quest is only accessible after you have completed Weapons On A Rampage. To access the graveyard you will need to buy the Graveyard Key from the Moogle Merchant in the children’s hideout, after you buy it head to the graveyard and kill the enemies within.
Kids On Patrol:
To complete this quest you will need to locate the 5 missing children, there is one near each exit for the town, one by the weapon shop, another by the community centre, and one by the tunnels.
A Verified Hero:
To unlock this quest you must first complete the Kids On Patrol quest. To complete this quest, you must complete the Whack A Box Minigame. After you complete this quest you will also now have access to the Pro Difficulty for the Minigame, completing the Pro version will also award the Crate Annihilator Acheivment.
The Angel Of The Slums:
To unlock this quest you must first do the Kids On Patrol quest. To complete this, you have to kill the monsters that the reporter ask you to kill.
Chapter 9:
BE CAREFUL WHAT DIALOGUE OPTIONS YOU SELECT AS THEY DETERMINE WHICH SET OF QUESTS YOU HAVE ACCESS TO.
To complete the group that you miss in your first playthrough, simply use chapter select in the menu to replay Chapter 9 and complete the requirements to unlock the other group.
For access to Group One, Party That Never Stops and A Dynamite Body make sure you select these options.
When asked about Tifa by Sam, select “She’s in great shape”.
When Sam Flips the coin select “heads”.
When talking to the man outside of Wall Market’s Hotel, select “No thanks”.
When you’re at Madame M’s Massage Parlor, select the Poor Man’s Course for 100 Gil.
When Aerith asks you what you think of her outfit, select “it’s alright”.
When you head down to the coliseum, drink the mystery drink on the table in the room just left of the entrance to the arena.
When you see Johnny running, follow him and complete his Discovery Event.
For access to Group Two, The Price Of Thievery and Shears' Counterattack make sure you select these options.
When Sam asks about Tifa, select “She’s great at handling the books”.
When Sam flips the coin, select “No deal”.
When talking to the man outside of the Wall Market Hotel, select “How much?”.
When at Madame M’s Massage Parlor, select the luxury course for 3000 G
When Aerith asks what you think of her outfit, select “it matters what I think?”.
Do not drink the mystery drink found in the coliseum.
Ignore Johnny’s Discovery Event.
Burning Thighs:
To complete this quest, you will have to complete the Squat Minigame in Jules’ Gym. This also unlocks the Pro difficulty for the Squat Minigame, which is required for the “Sultan Of Squat” Achievement.
GROUP ONE
Party That Never Stops:
For this quests you will have to go around reliving things around wall market to help an old alcoholic find inspiration.
A Dynamite Body:
This requires the completion of Party That Never Sleeps before you can access it. For this quest you will need to fight a match in the coliseum for Sam, select “Special Match” when you talk to the man in the coliseum.
GROUP TWO
The Price of Thievery:
For this quest you will have to go into the runners and kill the bandits.
Shears' Counterattack:
For this quest you have to talk to the man in the coliseum and select “special match” and complete the fight.
Chapter 14:
The Power Of Music:
This quest requires you to play 3 songs on the jukebox for Betty, they are “Goodnight, Until Tomorrow” found inside the Hotel’s lobby,“Stand Up” which is found in an alleyway to the left of the Honeybee Inn, and finally “Fight On” found in the coliseum.
Malicious Goons:
For this quest you will need to kill a Tonberry near Aerith’s house.
Wavering Heart:
For this quest you will have to beat Andrea in the pull up Minigame. Completing this will also unlock the Pro difficulty for the pull up Minigame, which is required for the “Peerees Of Pull-up” achievement.
Chocobo Search:
For this quest you will need to locate the 3 missing chocobos, the locations of the 3 are on your map and you will have to fight some enemies when you find 2 of the 3.
Missing Children:
For this quest, head to the graveyard and talk to Oates, this spawns some Ghosts, kill those and the quest will complete.
Tomboy Bandit:
For this quest you will need to complete another special match in the coliseum for Kyrie, complete the match and the quest will complete.
Subterranean Menace:
This quest requires you to go underground at Evergreen Park and defeat the Type-0 Behemoth enemy.
Secret Medicine:
This quest requires completion of the Subterranean Menace quest. For this quest you will need to gather three ingredients for the doctor’s medicine the ingredients are Medicinal flowers, which can be found in the church, Moogle’s Mortar, purchased from the Moogle Merchant, and Behemoth’s Horn, obtained during the Subterranean Menace quest.
Corneo's Secret Stash
This quest requires the completion of the Tomboy Bandit quest. This quest requires you to access all 3 of Don Corbel’s secret stashes, one is located in the Sector 5 Scrapyard, the second is located in the Abandoned Expressway, and the third is located in the sewers which are accessible in the main story path.
Also, if you complete all the quests available to you, you will acquire the “Devine Gratitude” achievement, nobody seems to know the exact requirements to get this achievement so this is probably the safest way to get it.