Spells

It would be nigh impossible for one to list every single 5e spell out there in a timely fashion, or to have to explain to a player why one spell in in a certain elemental category and another isn't.

To make this simpler, the author proposes a fix- reskin spells based on the player's element as well as the author's intent in each spell-element.

Admissible spells may be found on the DnD 5e wiki here, as well as this Reddit poster's brilliant conversion.

Description of Elemental Schools
Air. Air wizards control the air. They tend to focus less on offensive spells and have more than enough defensive and utility spells to cover for them. Spells such as Mage Hand and Gust are innately Air based, while spells like Psychic Scream or Slow could be reskinned to have an airy flavor. The appropriate damage types are all physical damages and Thunder. Inspiration: Aang from Avatar: The Last Airbender, Yoda from Star Wars, or Captain Planet from Captain Planet

Arcane. Arcane mages are rare, and tend to focus on divination and conjuration spells. Spells such as Eldritch Blast and Guidance are appropriate, as are more chaotic spells like Chromatic Orb. Spells like Lightning Lure can be reskinned for the Doctor Strange fans to get their Eldritch Whip on. The damage for Arcane spells can be all physical damages, Force, and Psychic. Inspiration: Albus Dumbledore from Harry Potter, Doctor Faustus from Faustus, or Doctor Strange from Marvel Comics

Earth. Earth wizards have a ton of defensive capabilities. They are able to control magnetic objects, raise pillars of stone, and cover themselves in hard sheets of granite for protection. The damage for Earth spells is solely physical. Inspiration: Magneto from the Marvel Comics, Hercules, or Roden from Buffy the Vampire Slayer

Fire. Similar to Lightning wizards, Fire mages don't believe in a strong defense. Optionally, the DM may allow Heroism and other buffing spells to be added, as in the author's opinion, Fire wizards should be a little bit more charismatic than the average Joe. Inspiration: Gandalf from The Lord of the Rings, The Wicked Witch from The Wizard of Oz, Kratos from God of War

Light. Light mages have a balance of utility and offense, able to control light and darkness in equal measure and focus beams into deadly strikes. While at first blush a spell like Disintegration should only be a Light spell. However, the damage for Light spells may only be Radiant. Inspiration: Revan from Star Wars, Raven from DC Comics, or Apollo

Lightning. Lightning wizards are all about the offense. They have few defensive capabilities, but can make use of the generous amounts of energy coursing through them to quicken their steps (Haste) and to help their allies recover (Lesser Restoration). The damage for Lightning Spells should be Lightning and Thunder. Inspiration: Emperor Palpatine from Star Wars, Azula from Avatar: The Last Airbender, the Energizer Bunny.

Nature. Nature mages control the forces of the natural world. They can befriend beasts, delve into necromancy, or become great healers. The damage for Nature spells should be all physical types and Necrotic. Inspiration: Radagast from The Lord of the Rings, Imhotep from The Mummy, or Jesus.

Water. Water mages control water, and perhaps even acid, blood, and other liquids. They are perfectly balanced (as all things should be), having spells like Shape Water and Tidal Wave alongside Acid Splash and Frostbite. Appropriate damage types are all physical damages, Cold, and Acid. Inspiration: The White Witch from The Chronicles of Narnia, Percy Jackson from Percy Jackson and the Olympians.

Reskinning Spells as a Ritual
If a player comes across a spell in a book that they are able to cast yet isn't of their element, they may create a new spell, using one of the appropriate damages of their element. This may require an Arcana check or other task, with increasing difficulty depending on the relation of the spell's native Element (for example, Lightning Lure being a Lightning spell)

Example. The player settles on being a Fire Wizard. He finds a spellbook containing Lightning Lure. After an Arcana check set by his DM, the player transforms Lightning Lure into Incendiary Whip, which functions the same way as Lightning Lure yet deals Fire damage. Since the nature of the spell isn't changed and Lightning has the same overall feel as Fire, the DM may not rule that the spell, a cantrip, is a higher level or any other additive.

Example #2. A different player- an Earth wizard, finds the same spellbook with Lightning Lure. After the check, he turns the spell into Magnetic Lure, describing it to only attract foes wearing metallic armor, and dealing them Bludgeoning damage as a result of being slightly crushed. Since the nature of the spell is slightly different, and Lightning a wholly different combat style to Earth, the DM may rule that it is a higher level, requires concentration, or simply a higher Arcana DC.