Scattered throughout the Kanto and Johto regions are move tutors. These NPCs will teach your Pokemon certain moves. The first time is always free.
There are also Egg Move Tutors. These NPCs will teach your Pokemon moves normally learned through breeding. Breeding does not exist in PokeOne. Egg Move Tutors require the use of Heart Scales which can be obtained from Rock Smashing rocks.
**REMOVING A HM MOVE WILL NOT WORK AND YOU WILL LOSE YOUR HEART SCALE**
Route 4: Mega Punch/Mega Kick
Rock Tunnel: Rock Slide
Celadon Dept. Store (3rd Floor): Counter
Silph Co. (2F, Juliet): Thunder Wave
Cinnabar Lab: Metronome
Viridian: Dream Eater
Victory Road: Double-Edge
Cinnabar Island: Egg Move Tutor
Fuchsia: Move Deleter (only way to delete HMs)
Saffron City: Move Reminder (Relearn old moves)
Cianwood: Egg Move Tutor
Move Reminder (Relearn old moves)
Move Deleter (only way to delete HMs)