After you defeat Elesa and get the Bolt Badge, you can start your journey to Driftveil City in the west.
Get Rid of Team Plasma Grunts
Go to the northwest part of Nimbasa after defeating Elesa. You find Hugh facing off with a bunch of Team Plasma dudes and dudettes. Talk to Hugh, then to any of the Team Plasma Grunts. There is Team Plasma Grunt ♂ who has a level 27 Trubbish. There is also Team Plasma Grunt ♀ who has a level 27 Watchog.
After the cowardly Team Plasma Grunts run away, we learn that five years ago, Team Plasma stole Hugh's little sister's Purrloin that had been given to her as a present. Hugh tells you to become stronger so you can help him fight Team Plasma. He heals your pokémon.
Go to Route 5
In route 5, you encounter Artist Horton in the grass. He sends out a level 25 Sigilyph. Behind him, there is a Super Repel.
In front of the trailer, Baker Jenn challenges you. She has a level 24 Sunkern and a level 24 Combee.
On the left side of the route you find the heartbreaker, Charles. He will challenge you to a rotation battle since you have four gym badges. He has a level 26 Archen, level 26, Sigilyph, and a level 26 Tirtouga. You have to defeat him to continue west.
After Charles rides away, you can battle Backpackers Ava and Aya. They have two level 24 Minccinos.
Exit the route to the west.
The route 5 gate takes you to Driftveil Drawbridge. Talk to the guy near the east side to get a new Funfest Mission.
While walking along the bridge, look for shadows on the ground, accompanied by the sound of wings flapping. If you step into one of these shadows, you might encounter a Ducklett. If not, you will get a feather.
Take the west exit of Driftveil Drawbridge to reach Driftveil City.