After you battle Team Plasma Grunts on the Plasma Frigate, Cheren tells you to go check out Route 6. You can explore Relic Passage first, or just go to Route 6 as described below.
Explore Route 6
The beginning of Route 6 is to the west of the Driftveil Gym. Cross the bridge. Then go north through the grass. The trainer here is PKMN Breeder April who sends out a level 27 Minccino and a level 27 Deerling.
Directly north of April is a hidden grotto. Check on the trees to find it. In the grotto, you might find Foongus, Nosepass, Woobat, or Eevee. The pokémon will be between level 25 and 30.
Go east across the bridge and fight Parasol Lady Nicole, who has a level 28 Castform.
Go north. Here you find School Kid Mara. She sends out a level 27 Deerling and a level 27 Shelmet. To the right of the school kid, there is an Elixir. The pokéball north of her is actually a level 29 Foongus.
Go west across the bridge. Cheren meets you there and takes you to the Weather Institute. He mentions that there was a mysterious sudden drop in temperature near Lacunosa town, Castelia and Driftveil. Cheren gives you HM03 Surf.
If you have an empty spot in your party, one of the scientists in the Weather Institute will give you a level 30 Deerling that has the hidden ability Serene Grace.
Outside, the scientist in front of the building (Scientist Jacques) has a level 28 Deerling.
In the fenced area next to the Weather Institute, the pokéball contains a Moon Stone.
To the left of Scientist Jacques, there is Scientist Marissa, who has a level 27 Shelmet and a level 27 Karrablast.
Go south into the fenced grassy area south of the scientists that you fought and use the Dowsing Machine to find a TinyMushroom. It's on the right side of the fenced area in the spot where there is no grass.
Go west from the Weather Institute and cross the bridge. The pokéball in the grass is actually a level 29 Foongus. To the left, there is School Kid Alvin, who has a level 27 Foongus and a level 27 Karrablast.
To the left of the school kid, there is an Ultra Ball. Go to the left from there and fight Parasol Lady Tihana. She has a level 28 Castform. The pokéball south of her is actually a level 29 Foongus.
Go north from the Parasol Lady and cross the bridge. Cobalion will appear, then will run away. Rood and an old man will tell you about Cobalion, Virizion, and Terrakion.
Use your Dowsing Machine north of the bridge to find a Protein. Go east from here. Go into the house and talk to the lady to get your pokémon healed up. Then talk to the girl to get a Shiny Stone.
You can go east and explore the dark grass, where you can fight PKMN Ranger Richard, who has a level 28 Whirlipede and a level 28 Foongus. Richard gives you a Pecha Berry if you win.
Your next stop is Chargestone Cave.