Shortly after completing the first Orphanage quest, part of the floor gets a crack in it, making it require strength to past.
After getting strength and the strength badge, a new area of the Railnet opens (With no wild Pokemon), allowing you to visit the inner part of Obsidia Ward and Yureyo Headquarters. After getting the key you unlock another part of the railnet: the puzzle room. The Puzzle room has a bunch or tracks and levers that you need to connect (without any obstructions) in order to open two doors, the one on the left leads to the Cave section of the Railnet, and the one on the Right leads to Chrysolia Pass. After you are done enter the train.
After obtaining the Amaranth Badge, you will be able to restore the railnet by donating 100,000 pokedollars at the Grand Hall. You will then be able to pickup all the remaining items inside the railnet at the railnet underground entrance. This also unlocks the train stations of different wards.
After obtaining the Geode Badge, you will unlock the basement of the Grand Hall. One of the scientists will request you to deal with the angry pokemon near the Byxbysion Wasteland. There will be two waves of battle, the first one being a single battle and the other two being a double battle.
Just as you enter the cave room, just above you there is a smash-able rock. Behind it, there is a man who fell from into the street from Obsidia Ward and couldn't get out. Upon talking to him he leaves to go to Obsidia Pokecenter. Go there and talk to him, and he will give you a department store sticker.