These are the ones I've personally done so far.

The Start of a Long Career
Start A Game.

Halfway There
Play through day 15

Hope Springs Eternal / Chilling Truth
Forgive Vit. Shift doesn't matter? Go bankrupt during days 10-19? He'll help ya out.

Truth Can Prevail
Trigger Vits line by dinner and forgive. Go bankrupt between day 10-21. Day 23 Cavidi will visit interrupting Lucky? Choose option 1.

An Unexpected Turn of Events
DON'T choose option 1 from Truth Can Prevail. Vit will end up having bank issues and ask for about 200,000 zunnics. If you give him the Zunnics, and don't win the tournament yourself, he'll win the F.L.A.B tournament and GIVE YOU the 1st place.

The Worst of Vices
This is for if you DON'T give him the money from the above mentioned achievement.

Settling The Score/Total Annihilation
Dinner shift, forgive Vit. Don't go bankrupts. Day 23 Vit starts the challenges. Don't tell Cavidi the truth about Gas's meal, I.E. no Meat Pie answer, choose one of the other two.

Mythical Beasts
This is just how I got it, I didn't get it when going down the Doctor's story line, but I did here, so . . . Make sure to tell Cavidi that Gas had a meat pie, he shows up and Lucky is a big mouth and we get penalized for betting.

Code Blue.
Ok, so there's poisoning and allergic reaction. I THINK this gets triggered either way, but if you choose day shift one happens if the evening shift another. If you don't get it choosing one, than choose the other.

A Lady's Heart.
Gas man get's sick and put in da hospital. You gotta go give the boi a pizza. A Mushroom za. BUT END THE DAY BETWEEN 2-6 to get the nurse? Bring dat Mushroom Za. IDK if you have to do a Evening Shift vs. Lunch Shift. I don't THINK it matters?

Down the Drain
Gas man goes to sup at Vit's crib. Get sick. DON'T forgive Vit. MAKE SURE YOU HAVE A SHROOM ZA in yo menu. MAKE SURE YO SHIFT DOES NOT END BETWEEN 2-6 PM. That's how you get the nurse, not the Doc. Bring it. BAM?

Nut Breaking Quack
A further step down the road from Down the Drain. Might have to make sure to do Evening Shift. Do that, than acquire the ingredient in the scandal on day 13. QUALITY IS IMPORTANT. IT'S RANDOM WHAT INGREDIENT.

Mozzarella needs to be A, Sausage B, Rocket C, and French Fries D.

On day 14, the Doc shows up. If you have the ingredient, and you made sure to Down the Drain, he'll penalize ya.

Repititia Non Iuvant/The Punisher

On Day 19, keep having the offending ingredient and the doc will show back up. Lucky will knocks his ass out, he'll penalize us again. BAM.

Frozen Escargot
Continuation from Repititia Non Iuvant/The Punisher. Lucky get's locked up because she whooped the Doc's Ass.For some reason this whole line DOES NOT count towards getting 1st place in F.L.A.B with a penalty. Maybe if you make sure to not pay the fine in time for a double penalty? Yet to be tested.

A Thorny Problem
A continuation of the aforementioned line, just keep going? You probably have to do the evening shift to make Gas get allergic instead of Arsenic Poisoning. Just have the doctor do his fake inspections, and maybe tell F.L.A.B it was the meat pie that Gas ate for lunch.

The Great Scandal
I THINK this ALWAYS happens? There's always an ingredient that starts having . . . problems, so probably can't miss this?

A Beastl (e) y Scandal
Again, I THINK this ALWAYS happens? I'm not sure.

Bad Things Come in Three
This one does NOT always happen. My current theory is you have to go to the Don line by doing a morning shift. It should happen as long as you continue down that path.

The Same Old Cliche
Aight, so a Piggy will show up and try to rob ya IF YOU DON"T GIVE THE DOCTER SHROOM ZA You have to play on a morning shift. If you play along with Lucky's silliness he . . . gonna try to shoot her. Yeah. That happened. I don't think the 2nd option triggers it, since he don't try to shoot her. The first option also gives a clue (?) about how to I.D. him with his 50 cal.

An Offer You Can't Refuse
DON'T REPORT THE PIGGY. He'll offer you to use his delivery service. Accept it.

Heroes by Chance.
Ok, another maybe complicated one. I'm not sure how this one triggers tbh, but you've got to try to identify the piggy. I got it by misidentifying him a little bit, with a 50 cal fedora Cuban identity.

Crime Has a Name
Perfectly Identify the piggy when you go to report him. 50 cal gun, Fedora, and the OTHER cigar, not the Cuban. Not sure off the top of my head.

Ubi Concordea, Ibi Victoria
Don't forgive him. Try to get the Noble Customer. Look at the Fit for a Prince in the recipes section. Before day 21? Report the piggy. When you chase that boi to the airport, if you get the Noble boi to show up twice I.E. get the pop up from him, a cut-scene should play? To be on the Safe Side, don't buy/sell the disco-ball. I guess this can piss Lucky off.

Checkered Flag
Just complete the story. One way, or another.

Towards New Horizons.
I THINK all you have to do is win that comp for them FLAB folks, but I see a achievement for getting Lucky locked up? Maybe don't do that.

An Unpleasant Guest.
Get 4 or 5 in the FLAB comp, be a grump. I know, it feels BAD, but you gotta do it for the achievements.

Aight, now this is not what I've confirmed, but what I've found just going through the discussions and trying to figure out.
MOST of this is from the DEVELOPER so it SHOULD be right?!? I gotta find time to play some more.

So it seems there's some key story branching areas.
There's forgiving and not forgiving Vit.
There's reporting the robber.
There's the quack in the hospital, and arriving to the hospital at the right time as for getting the doc or the nurse, depending who's on schedule.
There's even a thing for if you go bankrupt, so you might have to play on hard or PURPOSELY crash yourself??


False Advertising
You got to go to Grater Chef? You got to fail a bit, than the turtle show up. First option?


What Goes Around Comes Around
Lunch shift, don't forgive Vit. Refuse the piggy? I didn't see an option to refuse to pay him tbh. 5 days later when he come to rob you again, be too poor to have that much moneyz. Make sure you have a Disco-ball. It gonna get broke. Leave it broke? On day 26 I guess Lucky Breaks?

Piecing the Puzzle Together
Lunch shift, don't forgive? Gotta go on a Doc shift, unhealthy pizza/mushroom pizza. Day 25. Cavidi will ask you about Gas boi's din. Choose the 2nd or 3rd option? NOT the first? TBH, I feel like I did this. Option two was the Meat Pie? I choose the meat pie that day. IDK.

How to Start Grater Chef?
Choose Dinner, don't forgive Vit, take a NON MUSHROOM pizza on da Nurse Shift. That gets the nurse friends to come? Get a B, A, than an S personal score during a 12 hr shift?

This is NOT from the Developer like the other sections kinda were. So, grain of salt!
That said a LOT of this is from hirekatsu. I just wanted to throw this in here to be as through as possible. The ones with the achievement name next to it are ones I have verified myself.

Spring Pie Allegro Non Tropo Achievement.
Traditional Dough
Tomato Sauce
Fresh Tomato
Olives
Oregano
Anchovies

Simo
-Traditional Dough
-Feta Cheese
-Mushrooms
-Olives
All highest quality ingredients

Dr. Corvigna
-Traditional dough (high quality, but the others aren't)
-Parsley
-Oregano
-Lettuce
-Avocado
-Spinach

Lucky's Lucky The Gastropod Achievement.
-Traditional dough
-Tomato sauce
-Basil
-Eggplant
-Parmesan
-Mozzarella

Gastone's Making the Boss Happy Achievement.
-Traditional dough
-Tomato sauce
-Mozzarella
-Bacon
-Egg
-Peas

De Cavidi's Experimenting With Flavours Achievement.
-Rye dough
-Tomato sauce
-Mozzarella
-Onion
-Smoked reindeer
-Mushroom
-Oregano

Alpi's Turkish Pepper Achievement.
-Rye dough
-Fresh tomato
-Rocket
-Onion

Boe's Zeta Reticuli Achievement.
-Traditional dough
-Mozzarella
-Provolone
-Sausage
-Spinach

Strafy's The Devil Wears Camo Achievement.
-Traditional dough
-Basil
-Chedder
-Parsley
-Pepperoni
-Sausage

Brush's Still Life With Pizza Achievement.
-Traditional dough
-Dill
-Lettuce
-Pear
-Radicchio

Basilio's The Pressure Goes Up Achievement.
-Traditional dough
-Pesto
-Pecorino
-Parmesan
-Bacon
-Ham
-Salami
-Anchovy
-French fries

Michael's favourite. Fit For A Prince.
-Rye dough
-Olive p
-Brie
-Smoked reindeer
-Squid
-Asparagus
-Radicchio
-Pear

Angelino's favourite. Pizza Bombastic Achievement.
-Meat and Cheese.
(in my case)
-Traditional dough
-Ricotta
-All cheese ingredients
-Bacon
-Salami
-Sausage

Super balanced pie. Umami? Oh Mama! Achievment.
-Rye dough
-Pepperoni
-Chedder
-Rocket
-Spinach
-Avocado
-Pineapple

Vit's
-Traditional dough
-Mozzarella
-Gorgonzola
-Pear
-Walnut

Paying the Pizza in Pizza
Get in Big Trouble with the Famia? Maybe let piggy go without ID'ing him right? Maybe get bankrupts and saved by the Familia? IDK.

Trying Your Luck.
Play the Lottery? IDK

The Truth is Hard to Swallow

Cat Jerky
Just . . . fail?

A New Career

The Last Shall Be First?
Wooden Fork?