Week 1 Event: Orientation
Vayne meets Jess, who takes him on a quick tour of the campus. Visit the infirmary, student store, and student affairs before a cutscene appears and the two are whisked off to the workshop by Flay.
Save as soon as the Save tutorial is finished. It takes approximately 12 minutes to get to this point without reading most of the text on the PSP version. It's a good place to start a new game if you ever feel like starting over, so you might want to hold onto that particular save file.
You will meet Nikki and the Vice Principal at the workshop, and then receive a quick tutorial on save points. Flay puts the three of you to work making a Nicro Cloth in two weeks in order to save the workshop for yourselves.
Week 2 Class: Synthesis I Basics
For now, there is only one course available, and it's required, so you may as well accept it.
Your first assignment is to collect a Clearwater, Blue Petal, and Spinacherb. Clearwater can be obtained from the faucet in the Workshop, while the other two may be found in the Living Forest. After gathering the ingredients, return to the Faculty Room.
At this point, Off Campus becomes available, and this is where you'll find the Living Forest. The Search and Monster tutorials will come up right away, followed by the Battle Basics tutorial.
Wander the area cutting grass and fighting monsters until you've found the two ingredients. At that point you can either return to get the Clearwater and finish the assignment, or continue looking around to find more materials that may be useful later.
Visiting the Campus Grounds brings up a cutscene with Flay.
Return to the Faculty Room with all of the ingredients to receive the recipe for Healing Medicine (which allows you to make Healing Pot). You'll get a quick tutorial on synthesizing and produce a Heal Jar. Once synthesis is complete, the assignment is finished.
Week 3 Class: Combat I Basics
When ready, go to Student Affairs to sign up for another class. You still don't really have a choice on what class to take, so just accept Combat I Basics for now.
The assignment for this class is to defeat a Platinum Puni in Wind's Corridor. You are expected to return with a Shiny Puniball as proof of victory. Return to Student Affairs when you have the required item.
Jess suggests visiting the Athanor Room to power up your weapons and armor before heading out. Head to the Workshop Hallway, the Athanor Room should be the first door you come to in the hall. Here you will receive the Athanor tutorial and learn to make armor, weapons, and equipment. You receive the recipe for Old Collar, as well as the ingredients Woodchip, Legien Steel, and Dirt. After synthesizing a ragged collar, you'll get the Grow Book tutorial.
At this point, Wind's Corridor will be available Off Campus. Head there when ready to attempt the assignment. In order to get the Shiny Puniball, you must fight the first boss battle, which will also be a chain fight (meaning you will fight several waves of monsters. The best way to do this quickly is to avoid all other battles and only stray from the path to grab chests and other items. The fight should be fairly easy if you get there without any earlier battles and get
Healing Echo up as soon as possible once one of the characters takes damage. The Platinum Puni should only take a few attacks to kill. Analyze is useful not only to identify new enemies, but also because it gives a boost to Vayne's damage output early on. Save health items for situations in which Healing Echo won't be able to save a character.
Week 4 Event
Flay arrives to save the day, sort of... You receive the recipe for Generic Cloth, which allows you to make the Nicro Cloth you need to save the workshop. However, you have to go out and collect ingredients, and Flay isn't going to help you here. The ingredient you need is a Tuft, and apparently these are dropped by a Lore Beast living in the Old Schoolhouse. At this point the Old Schoolhouse is open (Off Campus).
There's a save point on the second floor of the Old Schoolhouse, near the destination. There is a door next to the save point that leads to a room full of crates, barrels, and chests. There are some useful items in here, but nothing that's going to be missed in the long term if you decide to bypass the room. Up ahead you'll find the room containing the Lore Beast. Thankfully it's alone, but you'll still need to keep Healing Echo up and use everything you've got to defeat it. Once defeated you receive the Tuft (and possibly a recipe for Flame).