From Bulbapedia, the community-driven Pokémon encyclopedia.
Now here we're gonna ignore every Trainer we possibly can, except ones with Shadow Pokémon and unavoidable ones. Head right, down the stairs, to the far right, and up. Your first battle is as follows.
Level 33 Bagon
Level 32 Numel
Level 33 Shadow Meditite
Espeon's Confusion should OHKO Bagon, Numel should go down in a pinch, and Meditite has a very high capture rate. After the battle, head up the hall and down the stairs. You're on a bridge. There's a female who wants to battle you here.
Level 34 Mareep
Level 34 Cacnea
Level 33 Shadow Dunsparce
Dunsparce is also very easy to catch, Mareep and Cacnea shouldn't be too much trouble for you. After the battle, keep heading left. Under the stairs are a healing pad and a computer. Here you can save+heal. Now go up the stairs. Head upwards, avoiding the first Trainer, and heads towards the stairs. Another Trainer is in the way, but this one has a Shadow Pokémon.
Level 31 Lotad
Level 32 Beldum
Level 32 Lombre
Level 33 Shadow Swablu
Swablu should be extremely easy to catch, and all of his other Pokémon go down easy. but still get it into the yellow so as not to waste a Great Ball. After the battle, head downstairs, unless you think you should save+heal, because next up is Miror B. Beware of the other Trainer here, though. Once downstairs, it's just a simple hallway until you reach Miror B.'s room.
Cipher Admin: Miror B.
Awesome music starts when you enter his room. Walk up to Miror B., watch him dance. He notices you, and tells them to stop the music (awww). After a short speech, he challenges you. His music has already started up again. Yeah!
Cipher Admin Miror B.
Level 28 Ludicolo
Level 29 Ludicolo
Level 31 Ludicolo
Level 30 Ludicolo
Level 35 Shadow Sudowoodo
You better have a lot of Great Balls, because Sudowoodo is probably the hardest Pokémon to catch so far. The four Ludicolo should go down two turns each, barely denting your health. They can really stall the battle though, so take them out quick.
Upon defeat, Miror B. and his four Ludicolo run off. Open the box that appears to get the TM Snatch. At the top-right of the new room is a Macho Brace. Right below the Macho Brace is the Ein File P. In the room is also Plusle. Silva and Duking arrive just when you find Plusle. You're automatically brought back to Duking's house, bringing Plusle with you.
At his house, Duking will give you Plusle, a weak and useless Pokémon that better stay in your computer.
Plusle joined your party!
Level 13 Plusle
Duking will now be able to be in touch with you now that he has your P☆DA number. Leave the building. When you try to leave the city, the lady in front of the fortune-telling house speaks to you. Her name is Fateen. She tells you that something is going on up north. Rui notices this is where her grandfather lives, in Agate Village. Now you can exit the city and head north to Agate Village.
There's a Pokémon Center here where you can heal+save, and move around your party. From the Pokémon Center, go left, up, and down the bridge and towards the cave. The man outside won't let you in, so head back to the entrance of Agate Village, and to the third "floor" of the village. You will arrive at Rui's grandparents' house. You will engage in conversation with them. Rui begins to tell them about Shadow Pokémon, when the man who wouldn't let you in the cave appears. He and the grandfather leave because people pushed the man aside and rushed into the cave. Follow the two all the way down to the now-open cave.
A female Cipher Peon greets you and a battle begins immediately.
Cipher Peon Doven
Level 33 Spheal
Level 34 Carvanha
Way too easy, especially after recently fighting Miror B. Next up is a male Cipher Peon.
Cipher Peon Silton
Level 34 Shroomish
Level 34 Cacnea
You'll push that pushover over easily. Right before you leave the cave a female Peon appears from the sky. She thinks she scared you.
Cipher Peon Kass
Level 35 Baltoy
Level 35 Ralts
Level 35 Kirlia
Harder than the others, but only because she has three Pokémon.
Head forward, and you'll find a gray Cipher Peon and Rui's grandfather (named Eagun). You'll watch Myth Trainer Eagun get owned by Cipher Peon Skrub. Immediately after the battle this unique Cipher Peon challenges you!
Cipher Peon Skrub
Level 37 Wynaut
Level 35 Geodude
Level 36 Clamperl
Level 38 Shadow Hitmontop
You think he would've evolved the Wobbuffett and Geodude by now. Hitmontop is semi-hard to catch, Clamperl, Geodude, and Wynaut are a bit tougher than the Cave Path's Pokémon, but still not that hard.
Upon his defeat, you return home. Grandma Beluh starts looking for the Small Tablet you need, and Grandpa Eagun tells you to speak to Senilor. Senilor is the house just south of the Pokémon Center. You'll learn a bit of information, but head back to Eagun's house. By now Beluh has found it. On your way out, your P☆DA gets its first e-mail. You hear news that Mt. Battle is under attack by a Cipher Admin. Don't head there just yet.
Purifying Shadow Pokémon
Now here's how to purify.
Simply put the Pokémon you want to purify in your party. They must have lost all four bars of their Shadow Energy. You can get rid of it in many ways (calling out to them in battle, walking with them in your party, sending them out, etc.) Now go to the Relic Stone. Pick up the Ein File C Skrub left behind, and go up to the Stone. Now is a very good time to purify all the Shadow Pokémon ready for purifying.
One last note for Agate Village: There's a Name Rater in one of the houses that can nickname Espeon, Umbreon, Plusle, and purified Shadow Pokémon if need be.