The FNSM (Friendly Neighborhood Spider-Man) App is introduced relatively early in the story.
Swiping the touchpad opens it up and there will be 2 tabs: Activities and Crimes.
Complete all 14 objectives under the "activities" tab and the final objective unlocks (imo) a pretty rockin' reward and the trophy.
The objectives will require you to go all over the map but none of the challenges are hard and none of them have time limits. There's also a good variety of them so it's not repetitive.
The missions are all unmissable, can be done in any order, and the app updates as you progress the story.
I had the whole thing done before the main game's end:
LOOTERS
HARLEM TRAINS OUT OF SERVICE
ROBERS TARGET LOCAL BIZ
HARLEM F.E.A.S.T SHUT DOWN
WE'VE GOT A LEAD
MY TRUCK'S STUCK
F.E.A.S.T TOYS STOLEN
LOST CAT
CAN I GET A PHOTO
CAN'T FIND MY PIGEONS
ICE BREAKING MY CRANE
STUCK ON SKYSCRAPER
MY CAR'S GONE
EMERGENCY COMMS DOWN
THANK YOU, SPIDER-MAN! <- this challenge unlocks after completing the missions above