Fate/Zero high session 3

Episode 10

Rin is attempting to help Tokiomi in his magic studies... Tokiomi seals the book that she snuck into, then sends his wife and Rin away to live somewhere else for their own safety...

Rin goes to try and investigate the missing children... senses a servant's presence with her mana detector... not knowing any better, she goes after Caster's master, and actually manages to destroy his bracelet and free the children he captured...

Then as she's running away, her mana detector goes crazy as Caster's demons ambush her... then she gets rescued by Berserker and Kariya. Kirei is spying on Kariya this whole time...

Fate/Zero high session 2

Episode 5

Rider offers Saber and Lancer to join his army. After they refuse, Lancer's master Kayneth shows up, and starts mocking Rider's master Waver... Rider then eviscerates Kayneth, calls him a coward for hiding in the shadows and making Lancer do all the work...

Then Archer and Berserker both show up and start fighting each other. Berserker uses his noble phantasm "For Someone's Glory", which makes him shrouded in fog... then Archer uses his own noble phantasm, the Gate of Babylon, and summons a bunch of weapons from his portals to attack Berserker... but it doesn't work... Berserker just catches the weapons and uses them against him. Meanwhile, Kirei relays Assassin's intel to Tokiomi, who then uses a command seal to force Archer to withdraw and prevent him from revealing more of his identity...

Fate/Zero high session 1

This series has a lot of spoilers for the visual novel Fate/Stay Night, but it also can stand alone and it introduces the story and the concept of masters/servants and other magecraft stuff just as well as the VN does... I would still suggest reading through the entirety of F/SN before watching this series though.

Episode 1

A bunch of things happen in this first episode.
It shows when Ilyasviel was first born, in the Einzburn Castle hidden in the forest... Emiya Kiritsugu tells his wife Irisviel that he's worried about Ilya getting involved in the Holy Grail War...
Kirei is introduced, and is chosen as a participant in the next Holy Grail War... Kirei's father, a priest in charge of overseeing the war, is colluding with Tokiomi Tohsaka, another Holy Grail War participant.
Kariya Matou is introduced... He's a friend of Aoi Tohsaka, Tokiomi's wife, and her daughters Rin and Sakura... Kariya learns that Sakura has been adopted into the Matou family and his father is doing a bunch of horrible things to this girl to increase her magical powers... Kariya's father Zouken is obsessed with trying to win the Holy Grail, and Kariya tries to save Sakura by agreeing to take the worm injections and win the Holy Grail for him instead of using Sakura...

Two other mage characters introduced, Kayneth and Waver... Kayneth is a natural prodigy at magecraft, and acts really harsh and mean towards his student Waver who thinks that hard work and practice matters more than bloodline... Waver steals an artifact from Kayneth to try and summon a servant and prove him wrong...

Kirei summons his servant, Assassin. Then it shows Kiritsugu, Tokiomi, Kariya, and Waver each summoning their servants... Kiritsugu and Irisviel are introduced to Saber.
Kirei... he goes from one path to the next, never completing anything... Kiritsugu wonders what's driving this person and what his goals are...
Kiritsugu... he focuses on killing mages and nothing else... Kirei is puzzled by his actions... neither of them know what the other could want the Holy Grail for...