ok so first of all i have to explain how japan does religion because my reading necessitates a lot of caveats sorry.

basically, it's not the norm to be "religious", per se. many visit shrines, celebrate at festivals, pray at temples, etc, but most will not really admit that they "believe" in the true existence of gods. there's a saying: "born shinto, married christian, died buddhist" -- in reference to the fact that most do not subscribe to a single religion. the use of religious theming in works is affected by this lack of devout "pickiness".

when buddhist theming is invoked despite the fact the togawas aren't card-carrying devotees, think of it in the sense of when christian symbolism is used in ave mujica stuff. it's important flavor, yes, but moreso a cultural callback to help with the message rather than "this is what the characters believe in, in-story"

therefore it's not even background really (because to me that implies there is a Tangible Solid Intent), it's like... stray symbolism the writers pick out, that when read a certain way, seems to paint a fun picture. it's not necessarily what the writers set out to portray. does that make sense?

WITH THAT OUT OF THE WAY.

i don't think this is... well, it IS very recommended, i just don't think most people would want to do it: peruse the pali canon. that's what i do sometimes in my spare time... haha... this is where i read, https://www.dhammatalks.org/suttas/. the dhammapada is of interest because they are verses directly attributed to the buddha himself. so they are the most streamlined way in which you can learn to understand what buddhism is about.

i would also like to direct you to the majjhima nikāya because they're not really that long (lit. "middle-length discourses") and it's generally pointed to as the place where the buddha sometimes drops in actual autobiographical accounts that hold water.

it's hard to pick out a "complete" narrative of the buddha's life (which i like to JOKE is what saki's life is invoking, esp the great renunciation... and also another thing, down below) but a cursory google search can give you many summaries of the "conventional" narrative. i know of a novel, but i haven't read it yet.

with that baseline knowledge established, we can look into the naming conventions of the togawas.

togawa sadaharu (定治) - 定 is used as a buddhist term to refer to samadhi, the last of the noble eightfold path, basically "complete and total meditation", total concentration of the mind. more poetically, his whole name can be read as "the right/fixed way to rule" -- this is a theme, especially in terms of deliberating on "the wheel-turning king" who will unite the four realms and bring us closer to virtue. the buddha is the sage's equivalent of this being. that doesn't mean i think he is one. after all, many passages in the scripture are satire of royalty, so i think you could point to this as irony... that this "right way of ruling", lacking in virtue, cost him his family.

togawa kiyotsugu (清告) - "confession of integrity", very simple. and again, very ironic. life is hard for one who is sincere, clean in his pursuits. the challenge comes when one who is virtuous suffers from the cruelty of existence, because that's when evil can creep up on the mind. evil does not immediately set in to the mind (especially that of a then-virtuous fool), but that's why the downslope is hard to catch, and is horrible when it comes to a breaking point.

togawa mizuho (瑞穂) - an actual normal name!? thank god. it's like, "fresh ears of rice", so it was used as a poetic name for japan, "the land of abundant rice". i'm not letting you go yet though, because when you combine the first kanji of her name with sakiko's you get 瑞祥/祥瑞, which are interchangeably used to refer to "auspicious omens" -- like, legendary creatures, supernatural happenings, pure showings of good luck, etc. extending that, 祥瑞 often mark the birth of children destined for greatness. on that note...

togawa sakiko (祥子) - i've written about this already! buddhism, in a japanese context, is very much associated with death. the subject's a bit too broad for me to give any sources. i don't know. maybe i'll write a paper fr

for the more specific (and relevant to ave mujica) concepts you should look into immortal consciousness, the non-existence of the soul (anatta), and rebirth. i'll try to explain:

basically, there is no material (especially not static) "self" but we do have a sort of "essence" that is linearly consistent through every incarnation -- you cannot exactly define the boundaries of a river, for it is continuous and ever-moving, and all the water "substantiating" it will eventually pass through. the water at the head and the water at the mouth of the river have vastly different chemical compositions, yet both are still "the river". this consciousness, this "essence", is separate from our physical being in this realm, but as we polish our existence in this realm through ritual, that "essence" is closer to freeing itself from the yoke of reality via rebirth.

this is... basically the dolls' story, yes? yes! it's so fun to go back to the 1st/2nd/3rd trilogy with the general knowledge of this stuff in mind. please do it if you have the time :pray:

i'll bring up "important info" (im assuming this means their main anime storylines) first. it's not much

i would say only mygo's intermission at bandori 12th live is fundamental, i have it subbed here https://b23.tv/BA6sOFy. fun fact, it was even mentioned as part of the "watch order" in the movie pamphlets!

we have the manga coming https://comic-growl.com/episode/2550912964982649376 and the movies should... EVENTUALLY get on bluray. hopefully

for ave mujica, they have their intermissions here https://docs.google.com/spreadsheets/d/1DBK-rJRVUkEytYobidfWmVr178RDZri4raGyLJTrq7o/edit?gid=0#gid=0. everything up to 0th is its own storyline not involving the dolls

thats the "important info". nows the fun part...

mygo has individual character covers! which are supposed to Say Something about who sings them. mostly they're tomori songs. and she's sick in the head. but i like taki's shiori and raana's teenage riot for characterization too https://youtube.com/playlist?list=PLUFFl4hYd1R3DbWVEUf1Nk6LUN8f-yjwX

they have short intro/QnA vids as well! i think its a fascinating relic from their earlier era tbh and its interesting to see how its aged https://youtube.com/playlist?list=PLUFFl4hYd1R1Pq4A_W2UmVbw9-wLCETFb

there's their weekly radio show, Maigo Center. it used to be purely in-character but they have seiyuu episodes now as well. a lot of the early episodes aren't subbed unfortunately but one day maybe(?) i'll get to that... https://youtube.com/playlist?list=PLUFFl4hYd1R0SQPSCAqPVuT5F2By4pqO7

in garupa, so far they have two unit events and two mixed events
breathing next to you (taki+raana) https://bestdori.com/community/stories/130897/
the colors muddled with the raindrops (soyo+tomori with some taki) https://https://bestdori.com/community/stories/144007
sakura circring (mygo meets the rest of the cast) https://twitter.com/dewdroppes/status/1769358187260428604
the scent of our blend, wafting by the window (taki works at hazawa coffee for a day) https://bestdori.com/community/stories/170215

i have their bday cards translated in my directory https://tinyurl.com/dimadirectory
...i realized i dont have anon's bday story linked there. whoops lol https://twitter.com/maigosusume/status/1833004852001259698

there's also three dreamfes stories (two upcoming)
raana df (untranslated? afaik)
soyo df (upcoming)
tomori df (upcoming)
they're basically two-part character studies

for avemuji...
refer to the masterlist above.
read the mujicarni article,
watch the lore videos,
watch prelude to 0th,
read the interviews,
listen to the songs and read the lyrics,

and welcome to the world of ave mujica :)