For this, first get the new jobs from the Shinjuku Hunter's Association and K's Tavern in Mikado. You can also get one from the northwestern section of the outer Shinjuku area (just above the underground) if you've done "Deliver the Film", which is a prerequisite for a number of later quests.
"Creating Fireproof Clothing:"
For this, you need to find four Kaso Hides and deliver them to K's Tavern. You might already have the four but, if not, they're found in the Kasumigaseki Counter-Demon Force Base. Note that the completion of this mission also makes the Fire Guard available at Q's Blacksmith.
For this, you need to deliver four Succulent Porks to a Hunter Association. These items are commonly dropped by Katakirauwa, found often outside Ueno's underground district.
"Phantom of Madness:"
For this particular quest, you'll find that some Insane Hordes are wandering around Shinjuku. There's too little to them to make them boss-worthy; just know that they're weak to Fire and enemy-party-wide attacks hit three times on them.
It will be available after finishing Corpse Disposal and the monsters in the area where you do said mission are found on the map (after finding the one in the northwest corner of the East Entrance: Main Street), so easy-peasy thatta way.
A demon boss, Chernobog, is fought just after the fourth Horde, but has nothing special about him and no true weaknesses; just an immunity to Dark.
This will be easy. Just go into the briefing room where all of the computers where in the Kasumigaseki Counter-Demon Force Base - you can get there via Terminal. Then just deliver it to the client.
"Training Battle 1:"
This is meant to teach you about the Reflection technique. Note that all Training Battles cannot cause a Game Over in any way. Anyways, when an element can be "Repel(led)", this means the damage is nullified on the target and turned back to the attacker or one of his allies.
Normally, you can't do this because of Flynn always being an open spot; see Upper Body Armor for my point. However, items or skills like Tetrakarn can Repel stuff. In this case, Gun and Physical attacks.
So, the solution? On Turn 1, use Tetrakarn and have everyone else just pass their turn. On Turn 2, just pass your turn. Loop between those two until you win.
"Training Battle 2:"
This battle is meant to teach about enemy weaknesses. Make Flynn use Lunge on Jack the Ripper, make Kaso pass his turn, then make Naga use Bufala on Pyro Jack, make Flynn pass his turn, then make Kaso use Maragi.