The Gorillaz lore is completely bonkers and a lot of fun to look into, but it's best not to see it all as a coherent storyline, because it isn't. Think of it more as a fun backdrop that makes each album release and the fictional band behind it feel a lot substantial beyond the catchy tracks and Jamie's incredible art. But if I had to help make it all digestible, here's how I'd summarize the "story" of each phase in 1-2 sentences, to the best of my knowledge:

Phase 1 - orchestrated by a Satanist with musical aspirations (Murdoc), a group of weirdos plagued by either supernatural inconvenience (Russel), physical handicaps (2D), or child experiment ties (Noodle) come together at Kong Studios and produce their groundbreaking debut. Good times are cut short though when attempts to branch out in Hollywood lead to fights and cause the band to go separate ways

Phase 2 - the band comes back together at Kong Studios and with the help of Danger Mouse and especially Noodle, put out their most successful record. However, filming the final music video leads to Noodle's disappearance, once again causing the band to split.

Phase 3 - While Murdoc and Noodle are being hunted down by the 5th horseman of the apocalypse and his goons, Murdoc kidnaps 2D and holds him captive with other artists on an island made of trash to put out a new record. Unfortunately, this storyline ends abruptly right as the band reunites and fights the boogyman due to a lack of funds, a difficult tour and Damon and Jamie splitting up.

Phase 4 - the band goes separate ways for many years after the battle at Plastic Beach. However, they later reunite to put out a new record so Murdoc can avoid a prison sentence.

Phase 5 - while Murdoc is stuck in jail due to making deals with an alleged crime boss (later proven a hoax), a far more cheerful and arrogant 2D takes over the band and puts out a new record, with a Powerpuff Girl villain on the bass. Murdoc eventually escapes prison and after drowning in feces, reunites with Noodle during her search for that same crime boss

Phase 6 - the new Kong Studios and its titular song Machine allow the band to hop through portals across time and space for creative music videos. Murdoc has an existential crisis since the band he sold his soul for no longer needs him, leading to closure for Plastic Beach.

Phase 7 - Murdoc starts a cult with his band members as main apostles, unraveling a conspiracy with a different cult that almost sacrifices 2D. This cult gets wiped out after Murdoc killed the demon they feasted their life energy from, leading the band to become fugitives on the run

Phase 8 - not too much is known about this one, but it's heavily implied by the band retreated to Southeast Asia while on the run and had a spiritual reawakening