Summarize:

You'll need HM01 Cut to get this one. You'll pass a man talking to his Poliwrath, then follow the road south and west again around a park. The buildings will disappear and trees will line the road, indicating that you're not in the city. As soon as you leave the city, you'll see a tree that's smaller than the others and next to a fence. Interact with the small tree to initiate the dialog box. Once the tree is cut down, you can walk through to the path behind it. You'll pass some kids waiting outside a house. You'll walk west through a hallway, past an old man, and through the other doorway to the back of the house. You'll see a girl and a Pidgeot. She'll ask you to keep the house a secret; say yes and as a reward, she'll give you HM02 Fly." "Fly" lets you use flying Pokémon to go to any Pokémon Center you have before.
Go to Celadon City. Head west onto Route 16 out of the city. Cut the tree down. Follow the path behind the fence. Enter the house and follow the path. Enter the second house. Talk to the girl.