There are new quests at K's Tavern, Shinjuku, Shibuya, and one from a girl in one of the rooms in the Tokyo Station, but that's it. You also get two new VR quests.
"Training Battle 7:"
In this order, use War Cry, War Cry, Diarahan, War Cry, Attack, Diarahan, and Attack 'till the end.
"Training Battle 8:"
Swap Ame no Uzume for the Tattooed Man, who uses Taunt. Put him back in the stock and use Next. On Turn 2, bring in Hel. On Turn 3, swap Hel for Pixie, who heals Flynn, then swap her back into the stock and use Next. On Turn 4, bring in Baphomet.
Swap him for Pixie on Turn 5, who uses Dia on Flynn, then later returns to the stock. On Turn 6, bring in Virture. Swap him on Turn 7 for Ame no Uzume. On Turn 8, use Sabbatama on Hel, then on Baphomet on Turn 9.
"The Deal in the Forest:"
From Mikado, head to the castle, then the city, then the Kiccigiorgi Forest. There, go east to the fork, south, then west and due west at the next fork into the "room". Speak with the man there of the quest and... Well, that was an easy five Summon Stones. However, you can get more by speaking with this man. Quests, I mean.
"Preparing for the Day to Come:"
You will need the "Bazooka Set" item for this. See Starting Investigation for more.
"A Magical Remedy" or "Magical Medication:"
If you finish one quest, the other is never to be completed again, due to plot-based circumstances. For the former, you need "Lung Medicine", for the latter you need "Euthanasia Serum".
Anyways, east of Ginza on the world map is the hospital. Defeat the Ghouls within - you've probably done this before; they're weak to Fire - to win some medicine.
"Escorting Your Comrade:"
First go to the end of the Kiccigiorgi Forest to meet the rather familiar man (heh), then simply go back to the main Mikado map. You can't go into the Terminals for now, so you will head into Naraku. Blaze through the lowest level - it shouldn't be hard - then to the overworld down in Tokyo.
You will meet up with some powerful demons at points on the way, such as on the 4th Stratum, but they're mostly just beefed-up enemies; nothing to worry about. After fighting the Kazfiel (weak to Fire), you can enter the Sky Terminal and go to Shinjuku.
From there, go to the area north of the Hunter's Association, then north another area, then to Fujiwara's place in the northeast corner.
"Take a Picture of Ginza for Me:"
The Luis Wilton store you must take a picture of is in Harumi Way B of the Ginza downtown area. See Sidequest: Downtown Ginza for more.
"Samurai School in Nakano:"
For this, head to the northwest of Shinjuku in the Tokyo overworld; pretty much the only usable area is the one in Nakano.
Marici is of little trouble to you; her attacks deal little more than 50 damage, and the only real elemental thing concerning her is the blocking of Fire. Just use those high-critical-rate moves to eliminate her ~3,100 HP and win.
"Save the Old Man:"
Head to Shibuya via the Terminals. There, go to the east, up the ledge, and east and outside. Head north there to meet the man; speak with him. But you'll be soon forced into a battle, albeit an easy one by now. Then another. The third fight is against Thor.
He resists Physicals, absorbs Electric, and is weak to Force; quite annoying when he has around 3,700 HP. He is rather difficult with numerous 100+ damage attacks, Megidola being party-wide and Mortal Jihad being a high-critical-rate.
"A Century of Defense:"
You won't find this 'till playing a New Game+. Head to Midtown in Roppongi again. Remember that vent on 1F near the elevator? Use it, then go through the door to the left to find and fight Take-Mikazuchi.
He is obviously weak to Force and drains Electric. Move-wise, he likes Berserker God (heavy Physical) and Ziodyne (heavy Electric). He won't last long, having maybe 8,000 HP, if you use Zandyne or Deadly Wind a lot. That'll do it, too.