From Bulbapedia, the community-driven Pokémon encyclopedia.
Upon entry you'll see a scene with a gullible cop and a thief/thug/no-one-knows-right-now. The gullible cop believes that the thug didn't do or take anything so he leaves him be. The thug snickers to himself.
Now get ready for a heck of a lot of Trainers. Use your usual strategies for battling normal Pokémon and Shadow Pokémon. Remember to use Makuhita and your Johto starter because their types and levels might be better for specific battles. You don't have to do all of these, but snagging every Pokémon requires you to skip all but the first two on the list:
One more thing, if at any time you need to heal, you can save and heal at the bottom-right area of the city in one of the buildings.
Level 25 Sentret
Level 25 Taillow
Level 25 Slakoth
Level 27 Zubat
Level 27 Gulpin
Pushover, and she doesn't even have any Shadow Pokémon.
Level 28 Skitty
Level 28 Zigzagoon
Level 30 Shadow Misdreavus
Makuhita is useful for killing here and Misdreavus shouldn't present a problem catching. Misdreavus will be a good asset to your team in the long run.
Bandana Guy Divel
Level 29 Psyduck
Level 30 Shadow Quagsire
Psyduck is a pushover if you take him out early. Go easy on the attacks, Quagsire is very weak and you don't want to kill it.
Roller Boy Lon
Level 29 Igglybuff
Level 28 Azurill
Level 29 Swablu
Level 30 Shadow Slugma
Don't use Water attacks on Slugma or he'll probably die. Igglybuff goes down fast, Azurill and Swablu shouldn't present a problem.
Level 27 Ledbya
Level 27 Wingull
Level 30 Shadow Noctowl
Ledyba and Noctowl can present real challenges. Focus on Shadow Noctowl and catching it, which is a feat in itself. Then Ledyba and Wingull should go down without too much of an HP loss on your part.
St. Performer Diogo
Level 26 Shroomish
Level 30 Shadow Flaaffy
Easy, though Flaaffy has a low capture rate, so have a few Great Balls handy.
Level 26 Oddish
Level 26 Dustox
Level 30 Shadow Skiploom
Skiploom should be easy to catch, and Oddish should be easy to kill. Should work well together.
After facing all of these trainers, go back and save+heal. You'll need it. If you're out of Great Balls at any point, you can go back to the Outskirt Stand with a little detour. Now you're gonna deal with that thug way over there, but first we've got to meet Duking and Marcia. Head left from where Roller Boy Lon was. Duking will tell you to leave him be, but the bookcase in front of the little kid on the left opens up. Enter and you'll find three children. The one on top is Marcia. She introduces herself and tells you to leave her be. The nerve of some people... Leave, and you'll find Duking talking to a boy with gray hair and the boy complains about Miror B. (the Cipher Admin from Phenac) but besides from one thing Duking is unfazed. The little boy turns around and tells you that his name is Silva.
Leave the room and go to the bottom of the city. Talk to the thug and he won't accept a fight. Go to the Pokémon Stadium/Colosseum at the top. Talk to the thug that isn't the thug we're looking for. He'll tell you the name of the thug is Cail (indirectly) and that if you win you get an awesome Pokémon but the match is going on right now. Now go all the way back to the bottom of the town. He'll accept your offer.
Level 29 Ralts
Level 29 Seedot
Level 29 Machop
Level 33 Shadow Furret
Furret breaks the chain of all-Shadow-Pokémon-being-Level-30. Oh well. Cail's levels have most likely surpassed yours unless you've exclusively been using your Eeveelutions Espeon and Umbreon. That's not a problem. Catching Furret relatively shouldn't give you too much trouble, but it will take a lot to weaken her down. Machop, Seedot, and Ralts shouldn't be a pain.
Save+heal after the battle. You can quit now, but there's still a bit more to do before you leave here for the first time.
The Gear and the Windmill
At the Shop buy a Super Potion or two. These come in handy. Head up town and you'll see a man in a lab coat walk out his door and fall. He asks for help. Just leave him lying and enter the building he came out of. You'll hear a brief word that Silva's gone too far and the Colosseum is out of power. You have to find Silva who has the Gear.
Silva has actually already left the city and is at the Construction Lot. Head there. At the top-right you'll see Silva has already left, but he left or hid the gear here. It's at the top-right corner, you can't miss it. Grab it and head back to Pyrite. Head all the way back to the windmill building. Duking is there when you arrive. Duking is a bit torn that Silva has betrayed his trust. Duking becomes fond of you two when he finds you have the Gear. Walk up to the gears and place the missing Gear. The Windmill is working again and the Colosseum now has power.
Right before you leave Duking asks for a little help because he thinks you're no "ordinary trainer". It's time to go upstairs and to the Colosseum!