Agate village is a small community town in Orre, but it has some important things for you to see in it. First off, you're heading for Eagun's house. His house is at the very top of the town in the back. Once inside, talk to Eagun's wife, Beluh, and Eagun will come inside and tell you to go to the Relic Stone.
The Relic Stone cave's entrance is slightly hidden; look left of the Pokémon Center to find a ramp that leads down to it.
Inside the cave are four (and another after) trainers that you have to battle. Just take them one at a time and run back to the Pokémon Center to heal between rounds. The fourth guy will give you a cologne case that holds scents; they can be bought only in Agate's Poké Mart and they lower your shadow Pokémon's Shadow bar.
At the Relic stone, Eagun will challenge you to a quick battle with his Pikachu. Use Grass type Pokémon because they resist Electric-type moves.
If you have any Shadow Pokémon with an empty shadow bar, you can click on the Relic Stone and have them restored to being normal. They will get all the experience points that they should have earned afterwards, so they will probably gain a level or two.
After you purify a Pokémon, Eagun will open the way to Mount Battle, one of the sidequests. If you're not ready for it yet, continue on to the Lab.