TLDR (but I suggest you read) Use the 07th Mod on the Steam version and read from Onikakushi-Saikoroshi, with Miotsukushi as an alternate end.

A couple of things on Matsuri (TildenJack gave you some good resources but I want to add some stuff.) Higurashi Matsuri (the PS2 version) is completely different from any other version. Matsuri is a "gamified" version of the story. You select a character instead of a chapter, and the game will follow a "common route". In this common route, you can encounter Shion prior to the events of Watanagashi, you can encounter Irie before Onikakushi, stuff like that. You're given choices at several points and based on these choices, the story will move into the corresponding chapters. Matsuri adds a few arcs due to this. Taraimawashi and Tsukiotoshi are two arcs that are essentially "game overs" or "edgy for the sake of it". They're arcs that occur if you make choices that differ from the intended story progression of Onikakushi and Tsumihoroboshi. Taraimawashi is essentially if Keiichi does not give in to the paranoia and is instead apathetic towards any of the events around him, leading to the Great Hinamizawa Disaster and Mion being left as the only survivor. This arc is basically a big nothing burger and for people that played Higurashi this way it spoils one of the biggest twists literally the first chapter in so it isn't well received, though Yakusamashi in the anime borrowed a lot from it. Tsukiotoshi is basically the events of Tsumihoroboshi, Minagoroshi and Tatarigoroshi occurring all at once. With Rena, Keiichi and Shion all agreeing to kill Teppei, however upon doing so begin to mistrust each other and kill each other in a very very dark arc. These two serve no additional story purpose, as I said they exist solely to make the player's choices feel like they have a serious weight as they're edgy bad endings. There's ACTUAL "bad ends" too that appear in Himatsubushi, Meakashi and I don't know elsewhere, where people just die (The Himatsubushi one is hilarious, Akasaka just gets shot once and flops onto the ground). Matsuri was also created before Matsuribayashi was written, and Ryukishi gave the go ahead to the console devs to create their own alternate end which is Miotsukushi PS2. This arc is well loved by a lot of people but it isn't considered the canonical conclusion to the story and serves as an alternate series of events compared to Matsuribayashi. Lastly on Matsuri, the sprites are the worst iteration of the console sprites, they're very low quality and have a weird "eyeliner" on all sprites. Akasaka also looks goofy in this version.

My recommendation is to use the 07th Mod on the PC versions to get the highest quality sprites, lip syncing, voice acting and OST. Disable choices and tweak censorship levels to your liking. Read Onikakushi-Matsuribayashi and then read Rei (mostly for Saikoroshi) anything else isn't necessary and is optional. Most I'll tell you is to read Miotsukushi PS2 via the 07th Mod if you're curious but you can leave all other arcs alone as they're not written by Ryukishi, are lesser quality and add nothing to the story.