WARNING: DO NOT play the game in story mode, as it disables some achievements (including story-based ones, potentially).
These achievements are simply for playing through each episode, nothing special needs to be done for these.
The Great Ace Attorney: Adventures
Defence Debut - Complete Episode 1
Crash Course Candidate - Complete Episode 2
The Realm of the Reaper - Complete Episode 3
Landed in London - Complete Episode 4
Fleeting Farewell - Complete Episode 5
The Great Ace Attorney 2: Resolve
Susato's Success - Complete Episode 1
The Key of Knowledge - Complete Episode 2
The Demasking - Complete Episode 3
The Legendary Pair - Complete Episode 4
The Risen Resolve - Complete the Final Chapter
Ace Aficionado - Complete the extra Escapade adventures (Special Contents > Escapades)
Note for the multi-part achievements
You are able to see your progress (and which ones you may have missed) for "The Top Rung", "Herlockian", and "Ace of Spades" in the 'Accolades' section of 'Special Contents' in the main menu. This can make it much easier to track exactly what you need.
The Great Ace Attorney: Adventures
Prime Presser - Successfully press a witness during the first cross-examination. This is basically storyline, but if you just present evidence without pressing you will miss this.
Inconsistency Seer - Successfully present evidence during the first cross-examination (Basically storyline)
Healthy Curiosity - When you receive the medical report card in your Court Record, make sure to inspect every point of interest (on the front, on the back, and inside once you open it). You receive the card during Trial, Part 1 but you should be able to inspect the card at any point after Kazuma has told you to do so, in case you didn't look at it all.
Pillow Perfectionist - During Investigation, Part 3, you will be tasked with finding where Kazuma was looking during the second "Dance of Deduction". This will be the last decision of the episode. During this time, you can inspect the bed in the area. Even though you can see beds earlier in the case, you cannot inspect them at that time.
Hat Hunter - During Trial, Part 3, you will get this achievement by pressing the third statement of the testimony "Rebuttal to the Accusation". Make sure you question Magnus when he reacts.
Second-Rate Strings - During Investigation, Part 1: Sholmes' Suite, you'll be greeted by Iris Wilson. Before conversing with her, examine the room and scroll to the left to inspect the violin. You must do this before talking about the 'Wilson' topic, or you won't get the achievement.
The Herlockian (1/7) - You must also inspect the blackboard on the right side of the scene before talking about the 'Wilson' topic, (same time as when you earn Second-Rate Strings), and you will read that Iris is working on something called 'Black Peter' (but she won't talk to you about it, yet).
The Herlockian (2/7) - After you talk to Iris about the 'Wilson' topic, inspect the blackboard again and Iris will talk to you about 'Black Peter'.
Ace of Spades (1/8) - You start the episode in Naruhodo's Legal Consultancy, and you can inspect a shovel (not a spade, as Naruhodo is told by Susato).
The Woman...? - Upon starting Episode 5, you can move straight to Sholmes's Suite and inspect the fireplace after the initial dialogue ends. When Susato finishes talking about the fireplace, the achievement will pop.
The Herlockian (3/7) - You can inspect Iris' blackboard right after earning The Woman...?, and you will converse with Susato about 'The Boscombe Valley Mystery'.
Ace of Spades (2/8) - At the start of Investigation, Part 3, you will be in Naruhodo's Legal Consultancy with Iris. Simply inspect the shovel the same way you did at the beginning of Investigation, Part 1 to be corrected about whether it is a shovel or a spade.
The Herlockian (4/7) - At the start of Investigation, Part 3 or 4 you can move around with Iris and not Susato. After talking to Detective Gregson, you can move to Sholmes's Suite to converse with Iris about 'The Boscombe Valley Mystery'.
Sulky's Story - During Trial, Part 1 (or theoretically any other testimony throughout the case where the Skulkin brothers are talking) simply press them on a statement and use the "pursue" feature on the brother that isn't talking. You have to do this once for each brother, I personally pursued Ringo on the 2nd statement of the first testimony and Nash on the 4th statement of the first testimony, but there are plenty more opportunities.
The Great Ace Attorney 2: Resolve
Father Frustrator - During Trial, Part 1 you will be asked to name the victim. Incorrectly claim that the victim is Taketsuchi Auchi (the prosecutor), then Yujin Mikotoba (your dad), and the achievement will pop.
The Herlockian (5/7) - As soon as you gain control in Sholmes's Suite, inspect the blackboard to talk to Iris about 'The Blue Carbuncle'.
Ace of Spades (3/8) - As soon as you talk to Iris for Herlockian progress, move to Naruhodo's Legal Consultancy and inspect the shovel.
The Herlockian (6/7) - After you get automatically moved to Miss Green's Hospital Bed for the flashback, go straight back to Sholmes's Suite and inspect the blackboard again for different dialogue with Iris.
Ace of Spades (4/8) - After investigating Mr Shamspeare's Room during the flashback, you'll be moved to Briar Road. Then, you can move back to Naruhodo's Legal Consultancy and talk to Susato about the shovel again. You can also have this conversation after the first part of the trial.
Down in One - During Investigation, Part 3 (after the first part of the trial), you'll go to talk to Miss Green after presenting the hidden photograph to John Garrideb. As you walk in to talk to Miss Green, you'll see her put down a bottle. Inspect this bottle for the achievement.
The Noodler - Near the end of Trial, Part 3, you will be asked for a sample to help locate Selden's loot. For this, choose to present a person, and then choose any person besides Olive Green (who is the correct answer).
Ace of Spades (5/8) - Once Herlock leaves his suite after talking to him, go to Naruhodo's Legal Consultancy and inspect the shovel to talk to Iris about it.
Inspirational Roots - When you are first able to go to the Prosecutor's Office, present your Armband to Lord van Zieks.
The Top Rung (1/2) - Once you can go to Madame Tusspells, scroll right to see the scene that Herlock is in. On the right side of his scene, there is a stepladder that you can talk to Iris about.
Ace of Spades (6/8) - At the beginning of Investigation, Part 2, you will be in Naruhodo's Legal Consultancy, simply inspect the shovel in your office again.
The Herlockian (7/7) - After talking to Susato and Iris at the beginning of Investigation, Part 2, move to Sholmes's Suite and inspect the blackboard to finally finish your Herlockian achievement!
The Top Rung (2/2) - After inspecting the blackboard, move straight to Madame Tusspells and inspect the same stepladder you did earlier with Iris to finish this acheivement!
Ace of Spades (7/8) - Upon entering Madame Tusspells Special Exhibitby examining the curtain after talking to Herlock w/ Susato in Investigation, Part 2, you'll see a wax figure with a shovel to inspect. You have to click on the shovel again when inspecting the figure in order to trigger the progress.
Ace of Spades (8/8) - After talking to Judge Jigoku and Professor Mikotoba at the start of Investigation, Part 1, go straight back to Naruhodo's Legal Consultancy and inspect the shovel for your final shovel/spade conversation!
Best Chips in Town - During Investigation, Part 2, you'll talk to Gina at the Fresno Street Room. After you lose consciousness and wake up in Naruhodo's Legal Consultancy, move back to the Fresno Street Room and talk to Gina.
Asogi Clan Appreciator - During Trial, Part 3, you'll get an update to the will with a poem on the third page. Then, inspect the hilt of the Great Sword 'Karuma' to find Klint van Zieks' will. After this, inspect the hilt once again for an achievement.
Bespoke Tailor - Play the second game with all of the alternate outfits. You can see a checklist in the 'Accolades' section of the 'Special Contents' menu. If you select all of the secondary costumes before launching the second game, this achievement will pop as soon as you start the second game.
The Great Ace Attorney - For obtaining all other achievements