Hogwarts Legacy

While playing through the story, you'll get various tasks from professors and upon completing them you'll get summoned to a class room or office and learn new spells. You have to perform the wandmotion once by pressing the button prompts for X, A, B or Y while following the spell motion.

You also have the option to learn the three unforgivable (dark green) curses, however refusing to do so does NOT lock you out of the achievement. Feel free to role-play your character how you want and choose the dialogue option to either learn the spell or not. The last one you need to learn is the transformation spell during winter season.

The other guide is correct, I just wanted to give a list of all the spells and where you learn them. It'll be obvious when you're in the game, as the quests list the spells you get as rewards.
For the majority of the spells, you have to complete assignments given to you by teachers as you progress through the game. For those you don't get this way, you'll learn them during main missions which are unavoidable.

Essential Spells

I think these spells are required, but you'll pick these up automatically as you progress through the story and they cannot be missed.

Basic Cast
You pick this up on the Path to Hogwarts quest, your very first quest in the game. Entirely unmissable.

Revelio
Same as above, you automatically unlock this as you progress through The Path to Hogwarts

Protego
The Path to Hogwarts again

Stupefy
The Path to Hogwarts again again

Ancient Magic/Ancient Magic Throw
You'll pick these up during the Welcome To Hogsmede mission, which is the fourth main mission you get. Before you can get this mission, you need to have completed Professor Ronen's assignment which unlocks the spell Reparo.

Alohomora
This unlocks about halfway through the storyline, during the mission The Caretaker's Lunar Lament. You need to have completed the earlier mission and be at least level 14 to be able to start this mission

Petrificus Totalus
You earn this during the Secrets of the Restricted Section mission, quite early on in the storyline. You need to be at least level 3 and need to have unlocked Incendio.

Control Spells (Yellow)

Arresto Momentum
You learn this during the second assignment for Madam Kogawa. This will be available a little while after completing her first Assignment.
You must fly through and pop the sets of balloons near Keenbridge Tower (the quest marker will take you here) and near the Spires (on the south side of Hogwarts). Once you return to Madam Kogawa, she'll teach you this spell.

Glacius
You learn this during the first assignment for the flying teacher Madam Kogawa. This will be available sometime after you complete The Map Chamber quest.
You have to pop the sets of balloons near Hogsmede Station (just southeast of Hogsmede) and around the Quidditch pitch. Once you return, you'll be taught this spell.

Levioso
You learn this during the fourth main mission Defence Against the Dark Arts Class.

Transformation
Along with the Unforgivable Curses, this will be one of the last few spells you learn. You unlock this after completing Professor Weasley's Assignment, which occurs about two-thirds of the way into the game. It's available to you after completing the quest Fire and Vice.
For the Assignment, you need to grab a couple of specific Field Guide pages, if you don't have them already. The first is in the Underground Harbour. Teleport to the Boathouse Floo Flame and swim out to the quest marker. You'll be able to spot a cave that opens into the Underground Harbour, then use Revelio to find the page.
The second is in the Library. Teleport to the Library Annex Floo Flame, enter the library go upstairs and find a student called Sophronia Franklin. She'll give you a little quiz about the Wizarding World. You don't need to get any of the questions correct, but if you fancy it the answers are

After the Quiz, you get the page after using Revelio.
With those in hand, attend Professor Weasley's Transfiguration class and speak to her once it ends, where you'll get this spell.

Force Spells (Purple)

Accio
You learn this during the mission Charms Class, the third main story mission and the first class you attend.

Depulso
To get Depulso, you must complete Professor Sharp's Assignment 1. You have to acquire and use a focus potion, as well as acquire and use a Maxima and an Edurus potion at the same time. If you are;t yet able to make these yourself, your best option is to go to Hogsmede and buy one of each potion from J.Pippin's Potions. Use all of them at once and you'll meet your requirements.
Attend a Potions class then at the end, speak to Professor Sharp to learn this.

Descendo
For Descendo, you need to complete Professor Onai's Assignment. This is available once you've completed Percival Rackham's Trial.
This requires a bit of fighting and knowing other spells. First up you have to defeat a Troll of any kind (you can find these in the Poidsear Coast area to the southeast, where there's a handful of troll lairs). Once you beat one, you have to collect the Troll Bogies that it drops.
You also have to hit a levitating enemy with Depulso. This can be done on any enemy you come across, Simply cast Levioso on an enemy (sure it's a human or gobin enemy, as these will actually levitate) and then cast Depulso on that same enemy.
Once done, attend a Divinations class and speak to the professor.

Flipendo
For Flipendo, you need to complete Professor Garlick's Assignment 2. This is available soon after you've completed her first assignment. You need to grow some Fluxweed, which you can buy seeds for from The Magic Neep in Hogsmede, then grow it in a large pot in the Room of Requirement (it takes 15 mins).
You also need to acquire the three combat plants, Chinese Chomping Cabbage, Mandrake and Venomous Tentacula, and then use all of them at once. If you're able to grow them, you can do that, but if not, you can buy them all from Dogweed and Deathcap in Hogsmede.
Once both of these are done, return to Professor Garlick.

Damage Spells (Red)

Confringo
You'll unlock Confringo during one of the main quests, In The Shadow of the Undercroft. Sebastian Sallow will teach it to you. You'll get this fairly early on in the game (in the first third).

Diffindo
Diffindo is learned from Professor Sharp's Assignment 2. Similar to the first assignment for Professor Sharp, you need to acquire and use two potions, the Invisibility potion and the Thunderbrew potion. You can buy both from J. Pippin's Potions, or brew them yourself in the Room of Requirement.
You can use the invisibility potion immediately, but the Thunderbrew potion must be used on enemies.
Once you've done that, speak to Professor Sharp.

Expelliarmus
One of the most famous spells from the Wizarding World, you learn this from Professor Hecat's Assignment 2. This is started after finishing Hecat's first assignment and the challenges are combat based.
You simply need to go into combat and dodge 10 times, This can be done in one fight or over multiple. You also need to cast Incindio on enemies 5 times. This can be 5 times on the same enemy or split over multiple uses and fights.
Once done, return to Professor Hecat in the Defence Against the Dark Arts classroom.

Bombarda
Another of the later spells in the game, you learn Bombarda from Professor Howin's Assignment which is available about halfway through the main story, after you finish the mission Charles Rookward's Trial.
After that, speak with Professor Howin and you'll have to go and capture two specific beats in the Nap-Sack. You'll need to nab a Diricawl and a Giant Purple Toad. For this, simply zoom all the way into the map and look for the paw print indicating an animal den.
Highlighting the icon will tell you what type of animal can be found there, simply look for the one you need.
For quick reference, there's a Giant Purple Toad den in the northeast of the map, just east of Upper Hogsfield. You'll find a Diricawl den just west of Keenbridge in the middle of the map.

Once you capture both, attend a Beasts class during the day and then speak to Professor Howin.

Incendio
You learn this from Professor Hecat's Assignment 1. This is unlocked very early in the game, just after finishing the quest The Locket's Secret. The assignments don't even need you to leave Hogwarts, you simply have to find Lucan Brattleby in the Clock Tower. Speak with him and organise using the training dummies he provides. Use the spells down on the screen to finish one practice. Then, have him organise a Duel with a fellow student. You need to win two rounds (one duel).
With both of these done, return to Professor Hecat.

Utility Spells (Blue)

Disillusionment
You learn this as a natural part of the main quest Secrets of the Restricted Section, also taught by Sebastian Sallow

Lumos
This is the only non-essential spell you learn on The Path to Hogwarts quest, the first main mission of the game.

Reparo
The first assignment that you receive on your first day at Hogwarts is to learn Reparo after completing Progessor Ronen's Assignment. You have to retrieve 2 Field Guide pages that are floating around nearby. Use the quest marker to track their rough locations and then look into the air to see a flying page. Use Accio on each page to collect them.
Once done, return to Professor Ronen.

Wingardium Leviosa
This is taught to you as a natural part of the main quest Charm's Class

Transfiguration Spells (Light Green)

Conjuring Spell
Both this and Evanesco are taught to you by Professor Weasley during the Room of Requirement main quest.

Altering Spell
You earn this from a side quest called Interior Decorating from Professor Weasley while inside the Room of Requirement. You'll need at least 15 moonstone for the first part of this quest, which can be found around the outside world by hitting the moonstone crystals with a basic cast.

Evanesco
Both this and the Conjuring Spell are taught to you by Professor Weasley during the Room of Requirement main quest.

Unforgivable Curses (Dark Green)

A note for these spells, you don't have to learn them for this achievement, but I've included them here for posterity.

Crucio
The first Unforgivable Curse that you can unlock, you can choose to learn Crucio during the course of the side mission In the Shadow of the Study. In order to unlock this mission, you must be at least level 16, and need to have completed the quest The High Keep, The Shadow of the Bloodline and at least one more quest after The High Keep.
You'll reach a part of that mission where you and Sebastian must make a choice. Either learn Crucio, and use it on Sebastian; Learn Crucio and have Sebastian use it on you or don't learn it, but still have Sebastian use it on you.

Imperio
Learned during the side quest In the Shadow of Time. You need to be at least level 17 and need to have finished In the Shadow of the Study.
During this quest, you'll once again be faced with a similar dialogue option to those in the Crucio quest, where you can choose to learn the spell or not.

Avada Kedavra
The final unforgivable curse and most infamous, you learn the killing curse during the side mission In the Shadow of the Relic. You must have completed the previous two curse quests and need to be at least level 28.
As with the other curses, during the course of this quest, you'll be given the option to learn Avada Kedavra or not. Be careful what choice you make and you'll come away knowing the killing curse.