After going west from Lavender Town and passing through Route 8 and 7, you reach Celadon City.
Explore the City
In the Pokémon Center, you will find Madam Celadon. You can pay her 10000 Pokédollars to set the Nature of the pokémon you encounter in the wild that day. For more information, check the Madam Celadon Guide.
A camper in the Pokémon Center, if you are playing Let's Go Eevee, will give you an Alolan Vulpix in exchange for a Vulpix, or if you are playing Let's Go Pikachu, will give you an Alolan Sandshrew in exchange for a Sandshrew.
If you south of the Celadon City Pokémon Center, you will find a tree that can be chopped down. Check the area to the right of that tree to find a hidden set of 5 Razz Berries.
There is a path behind the Pokémon Center. Go north on the path, then east, then south, to find a Nugget. Then go west on that path, then south, to find a hidden door. Go up all the stairs to reach the roof, then go west and into the door. Check the chalk board to read about Link Battles and the Communicate Menu. Next to the chalk board, there is TM44 Play Rough.
West of that hidden back door, there is a Rare Candy.
Brock is standing in front of the Celadon Condominiums in the north part of town. Walk up to him and there will be a cutscene. You will get some Pewter Crunchies from him. Then he will give you a cup of Tea.
In the Celadon Condominiums, you will find the Game Freak Development Office on the third floor. Check the water cooler to get a Fresh Water. On the top floor, there is a set of 3 Health Candies.
Celadon City Department Store
The Celadon City Department Store is in the northwest part of town. You can buy many things there, including TMs, evolution stones, clothing and accessories your partner pokémon, and drinks from the vending machines on the roof.
Talk to the guy behind the counter of the Video Game Floor to get TM03 Helping Hand.
Go to the roof and there will be a cut scene where you can buy a drink for your partner pokémon.
Talk to the little girl on the roof and give her a drink from the vending machine. If you give her a Fresh Water, you get TM06 Light Screen. If you give her a Soda Pop, you get TM09 Reflect. If you give her a Lemonade, you get TM07 Protect. She will still ask for a drink after you get all three TMs, but then she will change her mind and won't take the drink.
Find the Gym
In the southern part of Celadon City, there is a tree that you can chop down, then you can go south, then west, to reach the Celadon City Gym. If you are ready to challenge the gym, go inside and a Lass will ask to see a cute pokémon. Show her your partner Pikachu or Eevee, and she will let you in. Now you can challenge the Celadon City Gym and its Grass-type trainers!