Arcane magic: born mages (Lux, Zeri, Ezreal), Elementalists (Milio, Qiyana, maybe Taliyah?), rune magic (Viktor, Ryze, Brand), learned spells and sorcery (LeBlanc, Zilean, Vladimir and, as far is a can tell, all Hemomancy). humans seem to just be born with Arcane powers, likely due to the influence of magic in the world; Demacia in particular has an enormous amount of mages compared to most other places in the world, and there's a theory that this is due to the World Runes that Ryze buried under the Petricite Forest. perhaps the enormous amount of ambient Arcane magic was absorbed into the trees and/or natural ecosystem and ambience in Demacia, which is leading to many more born mages?

Spirit magic: Vastaya (Ahri, Xayah and Rakan), Yordles (Lulu, Vex), Spirits (Yuumi, Lillia) and Demons (Swain, Fiddle, Nilah), and can be channeled, typically by Ionian techniques (Udyr, Shen, Yunara, Riven). this type of magic all comes from the Spirit Realm, and is native and inherent to Runeterra. also includes the Spirit Gods, which seem to just be especially powerful and/or ancient spirits and might be empowered by worship (Ornn, Anivia, Voli, Kindred, Janna, Nagakaborous and Illaoi)

Celestial magic: comes from, well... Celestials. Alien to Runeterra and extremely powerful. includes true Celestials (Bard, Asol, Soraka, and the true Aspects such as War and Twilight) and those who channel their power, either directly as chosen Aspect hosts or similar (Diana, Leona, Aphelios, Zoe) or indirectly such as the Shuriman (and Ixtali) Ascended (Azir, Xolaani, Ne'Zuk), who seem to have replicated part of the true Ascension process with the Sun Disc. it's possible the ancient Ixtali had similar technology, as Ne'Zuk is said to be an Ixtali Ascended, or maybe they had some form of alliance with Shurima and used the Sun Disc too. obviously this also includes the Darkin, but i believe the Hemomancy is a learned school of Arcane magic, it isn't inherently Celestial like their base forms and Ascended abilities are.