From Bulbapedia, the community-driven Pokémon encyclopedia.
Mysterious Egg's status screen.
While I was battling trainers in the Azalea Gym, my egg just so happened to hatch. Depending on how many steps you have walked with your egg, it may hatch in here too. When it does, giveProfessor Elm a call and then head on over to his lab. I went ahead and picked up some repels because I wanted to get through the Union Cave quicker so I could get back to the gym. When you get there, Elm is pretty excited about the Pokémon who happened to hatch out of the egg, Togepi. He lets you raise this Pokémon and also gives you an Everstone. If you give the Everstone to your Pokémon, it will prevent it from evolving until you take the stone away.
After the Team Rocket business is over, you can challenge the Azalea Gym! Bugsy, the gym leader, specializes in Template:Type2 Pokémon. Bugsy's title is "The Walking Bug Pokémon Encyclopedia".
After defeating Bugsy you receive the Hive Badge well as TM49 Fury Cutter! You will also be allowed to use the HM move Cut outside of battle. When you leave the gym heal your Pokémon and then go to the Ilex Forest. Make sure you have a Pokémon able to learn the move Cut as well. You will receive the HM in the forest and you will need Cut to get through.
Your rival will challenge you to a battle before you enter the Ilex Forest. Here's his team:
If the player chose Chikorita:
If the player chose Cyndaquil:
If the player chose Totodile:
Once the battle is over go on in to Ilex Forest. Talk to the guy you see standing against the trees and he will tell you his boss's Farfetch'd got away. Go deeper into the forest and you will see the said Farfetch'd. If you go up to him and press A, he will say "Kwaa!" and run off. Keep doing this until you have chased him over to the kid you talked to previously. This time his boss is with him. Talk to his boss and as thanks for finding his Pokémon he gives you HM01 (Cut) Teach Cut to your Pokémon to get through the forest.
You will see a guy up against a tree in the forest. Go up to him and receive TM02 (Headbutt) This move enables your Pokémon to headbutt trees to find wild Pokémon. Usually these Pokémon are sleeping when you headbutt them out of the trees.
Once you leave the forest talk to the girl at the gate with the Butterfree. She will give you TM12 (Sweet Scent) Use this move to attract wild Pokémon.
Route 24 is home to the Pokémon Day Care Center, as well as some trainers eager to battle. Though it is a short route there is a policeman on duty here as well. If you talk to him at night he will challenge you to a battle.
Pokemon Day Care Center
Two people own and operate the Pokémon Day Care Center, an old man and an old woman. Talk to either one of them and they will raise your Pokémon. It costs 100 per level there. If you take your Pokémon before they gain a level you will pay 100 to get them out. In addition to raising level, you can also breed your Pokémon here to get Pokémon eggs. If you go in the pin where your Pokémon are at the daycare and talk to one of them, you will read one of three different messages depending on how compatible the two Pokémon are. If it says, "Its friendly with ---" you will get a Pokémon Egg quickly. If it says, "It shows interest in ---" then you won't get an egg as quickly. If it says, "It has no interest in ---" you will not get a Pokémon Egg. Those two Pokémon are not compatible. Another message is also seen if you try to breed Pokémon related to each other. (Brother and Sister, Mother and Son, etc.) This message reads, "Its brimming with energy." You will not get an egg if you see this message. The female Pokémon passes on the species so keep that in mind as you are breeding.
When you first arrive at the Pokémon Day Care Center, the old man will give you an Odd Egg. This egg will hatch in to one of the following baby Pokémon: Smoochum, Elekid, Cleffa, Igglybuff, Pichu, Magby, Tyrouge, or Togepi. This baby Pokémon will also know the move Dizzy Punch.