Speak to Alice, and you'll be entered into the Knock Out Festival.
Be careful travelling to Pao Pao Cafe after, as it'll immediately put you into a string of cutscenes and fights.

Reach Style Rank 9 with Terry, then speak to him.

You must finish the previous mission to enable this one.
Return to Beat Square and speak to Joe Higashi.

You must finish the previous mission to enable this one.
Return to Beat Square (yet again) and speak to Shopkeeper Biz.
Give the dog a flight ticket, then travel to Pao Pao cafe.

Reach Style Rank 18 with Terry, then speak to him.

After beating the first Knock Out Festival mission, return to Beat Square and check the poster beside your Hideout. Switch to Night and speak to Alice to enter the second set of fights.
After finishing, travel to Pao Pao Cafe again, then Aokigahara.

After finishing both Knock Out Festival missions, return to Beat Square and speak to Alice.

Reach Style Rank 9 with Mai, then speak to her.

Reach Style Rank 18 with Mai, then speak to her.

Note that you can only access most of the destinations required for this mission by first completing "Traveling with Mai!" and "Shiranui vs. Bushinryu!".
Go down a branching dock in Aokigahara, on the right and towards the screen, and you'll find a side path. Follow this path up to a statue, and make an offering of a Jaw Breaker Candy. Inside you can accept this mission to fight the five Ninjas within, each requiring a Ninja stone before fighting you:

• For an immediate stone, leave the cave, and speak to the Ninja just outside.
  
• Head to Beat Square (day), and there'll be another Ninja standing across the road from the subway entrance.
  
• On the left side of Fete Foraine, there is a child who will sell you a stone for 10,000 Zenny.
  
• On the far left of the Ranger's Hut, the old woman will give you a stone if you answer her quiz. The answers are Armadillo and Jaguar.
  
• Fight the guy on the left side of Enma's Hollow.

After beating all five Ninjas and completing the mission, you can choose to fight each one again to gain a Lyric card. Hand them to the man in the cave, and he'll give you a music track.

Enter Beat Square and speak to Anike. Travel to Kenya, then back to Beat Square, and talk to Narumi.

The elephant picture can only be taken in a certain spot, follow Houma's gaze, and walk a short distance away from him. The camera icon should appear once you're in the right place. The picture can also only be taken during the day.

Reach Style Rank 9 with Elena, then speak to her.

This seems to only become available after the previous mission.
Travel to Beat Square, run north towards the giant staircase, and you'll find Narumi.

Travel to the Metro City park at night. You'll know the mission is available as there'll be a dialogue line upon arriving. Speak to Lisa beside the statue.

You can get into the back area of the concrete... Structure... Thing... By dropping down to the platform with the chest on, then walking into the grassy area behind it.

Reach Style Rank 18 with Elena, then speak to her.