Here, I have have listed all of the HMs and the badges you need to use them out of battle. Again, ignore the words at the bottom of the page.

HM 01 contains CUT. If you have the CASCADEBADGE, you can use CUT outside of battles. It is found on the S.S. Anne, the captain gives it to you.
HM 02 contains FLY. If you have the THUNDERBADGE, you can use FLY outside of battles. In Celadon City, right before entering Cycling Road, CUT down the bush. Go into the house. A person inside will give it to you.
HM 03 contains SURF. If you have the SOULBADGE, you can use SURF outside of battles. If you get to the Secret House of the Safari Zone, a man will give you the HM 03.
HM 04 contains STRENGTH. If you have the RAINBOWBADGE, you can use STRENGTH outside of battles. Not far from the Safari Zone's Secret House, you will find GOLD TEETH. Give them to the warden in Fuchsia City to receive the HM 04.
HM 05 contains FLASH. If you have the BOULDERBADGE, you can use CUT outside of battles. After exiting Diglett's Cave, go south. You will wventually find the building where Prof. Oak's aide is. He will give you the HM 05.