Route 34 is a short route filled with trainers and is home to the Day Care Center. There are five different trainers, and if you visit at night, you can battle the policeman in the northern half of the route.
When you approach the Day Care Center, Lyra/Ethan will appear to visit their grandparents, who happen to be the owners of the Day Care. You will be invited inside by your friend to be introduced to the Day Care Lady. She will make fun of your relationship. You can now keep Pokémon here for them to raise!
When you exit, the Day Care Man will give you their numbers so you can be updated on your Pokémon's status from anywhere. Continue to Goldenrod City by traveling north. There is also an item in the last patch of grass located behind the Camper.