Wizard Progression

Brilliant scholars who use their knowledge to bend the very fabric of the universe to their own ends, wizards are mysterious masters of arcane magic. They are often mistrusted by those who cannot understand their thirst for knowledge. Preparing spells is incredibly difficult and sometimes dangerous for a wizard, but woe to any who would stand against a wizard who is prepared.
There are a number of wizard organizations in the world. These groups often impose strict regulations on local wizards. These regulations are usually enforced ruthlessly by the organization in order to maintain good relations with the local authorities.

The Short Version

Details are below, but here are the most important points to remember.

1. Starting when you pick your wizard school, you get one extra spell slot of each spell level (called your School Slots) that can only be used for spells associated with your school. These do not require a Preparation Check.
2. Spells you do not cast as a ritual take only an action to cast, but may take longer to have their full effects. (e.g. Control Weather would take 1 Action to cast, but the weather change would change over the course of the next 10 minutes).
3. You prepare spells into individual spell slots, and can prepare a given spell into as many spell slots as you like if you want to cast it multiple times. They stay in that spell slot until you finish a long rest.
4. You can leave a spell slot empty and then fill it in with a spell later. Doing so takes 10 minutes + 1 minute per spell slot level, during which you must Concentrate on your spell book.
5. When you prepare your spells for the day, you make an Intelligence (Arcana) Check (called your Preparation Check) for each spell level you know. Success refreshes the spell slots for that level, or even grants more slots, while failure reduces your spells of that level or might even cause a misfire.

Spellcasting

Wizards are unique in that they have no innate magical skills. Instead, they train their mind to hold pure magical energy in the form of complex equations and geometries known as "spells".
This manner of magic is by far the most difficult to perform, requiring intense study, memorization capabilities, and incredible discipline. In exchange for their intense study and years of dedication to the craft, wizards gain access to some of the most powerful magics known to Aedeva, and even beyond.

Learning Spells

At 1st level, you your spellbook contains one spell per Intelligence Bonus from your Field of Study (see below), and 2 spells of your choice from other schools, adhering to any restrictions from your Mage Guild (see below). You know the Tier 1 version of each of these spells. Higher Tier versions of spells you know count as separate spells and must be learned separately.
These are the only spells you learn intrinsically. All other spells must be learned from a Spell Scroll or Spellbook (see below) or from another wizard.

Learning From Scrolls

To learn a spell from a Scroll, you can spend 1 day per Scroll Tier studying the scroll. At the end of your studying, you make an Arcana (Wizardry) Check with DC equal to 15 + Double Scroll Tier (called a Learning Check). If you succeed, you copy the spell into your spellbook at the same Tier that it was placed into the scroll.
If you fail, you do not learn the spell. Regardless of success or failure, the spell is expended in the process.

Learning From Spellbooks

Learning a spell from a spellbook is exactly like learning one from a scroll, with two exceptions.

1. You cannot learn any spells from a spellbook until you decypher its contents, since all wizards encrypt their spellbooks, disguising them as cookbooks, poetry, stories, or similar. Decyphering a spellbook takes 1 day of study for every wizard level the previous owner had, and requires an Intelligence (Linguistics) [Secret Messages] Check with DC equal to the 8 + Previous Owner's Wizard Level.
2. Failing to learn a spell from a spellbook does not use up the spell or the spellbook. You can simply try again with the only cost being time and effort.

Learning From Tutors

Learning from another wizard is exactly like learning from a spellbook, but without the need for decryption. However, they usually require payment in some form or another (if monetary, it should be a payment of Tier at least equal to if not greater than the spell that is being learned).

Same Spell, Different Tier

When you learn a spell, you only learn it at a specific Tier. For example, learning Tier 3 Fireball does not allow you to cast Tier 1 Fireball, or Tier 6 Fireball, even if you are high enough level to cast Tier 6 spells. You must still learn the Tier 6 or Tier 1 versions of the spell as if they were separate spells.

Learning Higher Tier Versions of a Spell

This is exactly like learning a brand new spell with the following exceptions:

1. You have Advantage on the Learning Check.
2. When you gain a Wizard Level, you gain a number of Upgrade Points equal to Int Bonus + Roll of Proficiency Die. Each Upgrade Point can be used to learn the next Tier of a spell you already know, as long as you are high enough level to learn that Tier of spell.

Learning Lower Tier Versions of a Spell

This is exactly like learning a new spell with the following exceptions:

1. You can learn this Tier of the spell from your own spellbook (using your higher tier version of the spell as a reference).
2. You automatically succeed the Learning Check when your research is finished.
3. You can also use Upgrade Points from a level up to learn a lower Tier of a spell you know.

Daily Memory

Your Daily Memory is equal to your Intelligence Bonus + Proficiency Die (rolled at the end of each Long Rest). It determines how many spells you can have memorized at a time.
Starting at 10th level, your Daily Memory is equal to your Intelligence Score + Proficiency Die instead.

Memorizing Spells

To memorize a spell, you must have available Daily Memory equal to Spell Tier + Spell Class. Make an Intelligence (Arcana) [Wizardry] Check with DC equal to 12 + Memory Spent (called a Memorization Check).

Memorization Check Results

Crit Success: you memorize the spell. It uses 1d4 less Memory than normal.
Success: you memorize the spell and expend the Memory.
Fail: you do not memorize the spell but the Memory is still expended.
Crit Fail: as Fail, except the spell also runs wild. The effects can range from an explosion centered on you to the spell casting on you to a wave of wild magic, to anything in between. The GM is encouraged to have fun with this, but the side effects should never be positive, and their extremity should be related to the Tier of the spell being memorized. There is a reason that grey haired wizards are rare.

Memorizing a spell takes 1 minute per Spell Tier. Any number of memorizations can be done as part of a completed Night's Rest.

Casting Spells

Because wizards do most of their work during preparation, the actual casting of spells for wizards is surprisingly simple. Most wizard spells are cast as an Action. A few spells might have longer or shorter casting times, or delays before they take effect. In these cases, those details are specified in the spell description.

Components

If the spell requires any Verbal, Somatic, or Material Components, they are listed in the spell description. Such components are not strictly required, but are pneumonic devices to aid in the casting process. Casting without a needed component has a 3% chance per spell Tier to cause the spell to fizzle, wasting it entirely.
Unless otherwise stated, all wizard spells require line of sight to the target, and line of effect (which can be blocked by things like running water, magic circles, thresholds, and the like).

Ritual Spells

As a wizard, you can also learn arcane rituals. These are not spells, per se, but can have similar effects. They are learned in the same way as spells, but their casting time and component requirements are usually quite elaborate. However, their effects can be equally elaborate.