The warp zones are activated as soon as you reach the main hub screen of each area. These areas, in no particular order:
Blue Sky Zone, which you'll most likely discover first.
Graveyard Zone, which you can access by entering the lone cabin in the forest and exiting, then take the exit hanging under a bit.
Ruins of Zu Zone, instead of entering the cabin, you ascend up the tree and traverse past a large statue throne by entering the door at its base.
Industrial Zone, denoted by graffiti and signage. From Blue Sky, take the lighthouse exit, ascend through it by using a key, then enter the floating island. From coming out the other end, ascend to the water tower, repeat ascension, and you'll enter the new zone.
Sewer Zone, connected to Industrial. The door right under Industrial's warp leads to another spot. Once here, climb until you reach the top of the first tower and enter that door to find what Dot calls a well (but we know looks more like a pipe thanks to Nintendo). Proceeding through this will bring you to the warp cube and unlock the trophy for you.