HM 01 - Cut
Once you enter the Ilex Forest, you will encounter some men who are trying to round up the Farfetch'd that has gotten loose. Sneak up on the Farfetch'd to drive it back to it's trainer. He will reward you with HM 01 which is the move Cut; you can use this move to cut down small trees that block your path.
HM 02 - Fly
HM 02 allows you to teach Bird-type Pokemon to fly; this will save you time and energy when you need to get from one town to another quickly. Once you arrive in Cianwood, defeat the Gym Leader Chuck then go and speak with Chuck's wife; she will give you HM 02.
HM 03 - Surf
Defeat all five of the Eevee trainers who fight with the various evolutions of Eevee in Ecruteak City. Speak with their father when you have beaten them and he will give you HM 03, Surf, which allows you to surf across oceans and small bodies of water.
HM 04 - Strength
This HM does not require you to go on any special missions or side-quests. Go into the house next to the Pokemon Center in Olivine City and the woman inside will hand you HM 04, which contains the move Strength. Strength lets you move large boulders that block your path.
HM 05 - Flash
Before you get Flash you will have returned to Ecruteak City and will be able to access the Bellsprout Tower. Enter the tower and defeat the Monk of Bellsprout Tower; he will give you the HM 05: Flash. Flash lets you illuminate dark caves so you can navigate more easily.
HM 06 - Whirlpool
Help Lance defeat Team Rocket in Mahogany Town and he will give you HM 06, which is Whirlpool. This move lets you safely swim across whirling waters so you can access the Whirl Islands.
HM 07 - Waterfall
The last HM you will receive in the game is HM 07 which contains Waterfall. HM 07 is inside the Ice Cave, which you must go through on your way to Blackthorn City. Waterfall lets you swim up waterfalls. You will need this move to get to Victory Road and the Pokemon League.