HM is Hidden Machine
Unlike TM - Techincal Machine it nevers breaks
HM/TM teach your pokemon moves that are compatible with.
HM/TM can be found in the bag under HM/TM
Hidden Machine like certain TM can be used outside of battle
HM01 is cut which is the one you get in Azalea Town
To teach a pokemon an HM/TM go in bag and under the HM/TM catagory ( The one with CD picture)
You choose which one you want to teach your pokemon. Only certain pokemon can learn a certain HM/TM. After you choose what pokemon you want to use it on. It will act as if you learned a new move.
In Azalea town
You have to go to Kurt The Pokeball Creator. His house is in the top left corner of town, He has a white apricot Tree outside his house. You talk to him and he will go to the slowpoke well and you go there too. Just follow the path and you will fight team rocket grunts.After that you will automatically go back into Kurts house and he will make pokeballs from Apricots but they take a day to make. After that you go left out to the Ilex Forest. You will see a guy who has lost his farfetch'D. You kinda play like a mini- game trying to catch the Farfetch'D. After you get the Farfetch'D. The man will give you HM01 which is Cut
You teach HM01 to a pokemon and now you can cut down Trees Go through the Ilex forest and the game will tell you what to do
The Egg is a Togepi. It will hatch after a while
Hope I helped : D