From now on, the story splits into two versions: the Rin Route and the Rani Route. The actual steps to go through those are almost exactly the same, so I won't be writing them separately. From here on, anything specific to the Rin Route will be highlighted in red, and anything specific to the Rani Route will be highlighted in blue.
If you have Saber, go to your Private Room for an important conversation.
Go to the Library for an event with Leo.
You can leave and reenter the Library and talk to Leo if you want to learn more about the Holy Grail.
Go to the Nurse's Office and talk to Rin/Rani.
Talk to Sakura to get your weekly item.
Go to the 1F foyer for an event with Rani/Rin.
The pathway to the Trigger extends south from the middle of the passage along the north side. Go there and move down it until you pass near a shiny point and trigger an event. When the enemy Master appears, choose "Observe them."
You can't continue past this point, so explore the rest of the dungeon. You should be sure to pick up the Shockwave-2nd Strike item next to the exit in the northwest. It gets increasingly useful later in the game, so hold onto it.
You can continue past the point you couldn't yesterday, so do so and collect the 1st Trigger.
There is now a new Nephilim enemy type on the path right before the exit in the northwest. This enemy type is stronger than normal enemies and gives a lot of EXP. Also, if you're using Caster, defeating 5 of them will upgrade one of her Skills. This one doesn't regenerate after being killed, but you will have plenty of opportunity to hunt others in the future, so don't worry about it.
Talk to Taiga on 1F to accept the quest to get the Crab Omelet.
Go to the Arena Entrance for an event. You get Matrix Level 1.
Be sure to grab the Crab Omelet from the room on the east side.
Talk to Taiga on 1F to give her the Crab Omelet and accept another quest.
Go to the schoolyard and talk to the marked boy.
If you have Archer and are using a female character, go to your Private Room for an important conversation. Pick the only option and then either option 1 or 2 (they both get the same result).
Go to the 1F foyer for an event with Kotomine.
Your main goal is to defeat more Carnivore enemies than the enemy Master. They are represented by the larger-than-normal red squares in the map image. This is fairly easy to pull off; just run out and take out as many as you can. Although I would recommend heading toward the northern half of the map for easier pickings, and make sure you don't run into the enemy Master in the process. After all the Carnivores are dead, the enemy Master will leave.
Now, explore the dungeon normally at your leisure. Make sure you pick up the 2nd Trigger and Shark's Fin.
If you have Caster, go to your Private Room for an important conversation. Pick choice 2, then 1.
Go to the schoolyard and give the Shark's Fin to the boy.
Do the Carnivore hunting again like yesterday.
After that's done, the enemy Master will challenge you to a fight. You get Matrix Level 2.
If you have Archer, go to your Private Room for an important conversation.
Talk to Taiga on 1F.
Talk to Touko in the Chapel.
Go to the Arena entrance for an event with Kotomine. Assuming you won the hunting competition (which you surely should have), you get Matrix Level 3.
Enter Arena 2F and train until you're satisfied (and note this should be your first truly challenging boss fight, so whatever level of readiness you think you need, add some more to it).
Lancer starts the fight by buffing himself to strengthen his Guard counterattack. He doesn't really Guard that often (except for occasionally when he does nothing BUT Guard for an entire round), so it's not a huge deal. Lancer uses Ceremonial Purge for his 5th turn every 3 rounds, which does damage equal to how much you damage him, so be ready for it and ideally prevent him from using it. He always uses his Noble Phantasm on the 6th turn, and generally starts using it the 2nd round after his gold indicator starts showing, so definitely be ready to defend against that somehow (Guard doesn't work, so that means either Hack or a defensive Skill). He and Ronnie will converse in an ominous tone before the round when he uses it. Other than that, he's definitely easier than Berserker, so have fun.
Pretty straightforward fight. That's not to say it's easy, quite the opposite in fact, there's just no fancy cure-all strategy for it is all, so just fight. She only uses her Noble Phantasm on the 6th turn, which it should tell you when she's going to use it, just be sure to Hack or Guard it somehow or another, because it's pretty powerful. Note, just a heads up, she ALWAYS uses it on the 6th turn of the very first round. She Waits a lot, which gives you a substantial advantage, considering how high her attack power is. Remember you can always use an attack Skill on any turn you're not sure about, which will cancel any opponent attacks that are not Skills themselves, just keep in mind that it isn't easy to recover your Servant's MP (read: impossible at this point if you're not using Caster). And make sure you bring plenty of recovery items or equip a Formal Wear with a strong healing Code.