What gyms give you what badges, and what badges let you use what?

Here is what you receive from the Gyms (in this order):

Oreburgh City - Coal Badge - Can use HM Rock Smash outside of battle

Eterna City - Forest Badge - Can use HM Cut outside of battle

Veilstone City - Cobal Badge - Can use HM Fly outside of battle

Pastoria City - Fin Badge - Can use HM Defog outside of battle

Hearthome City - Relic Badge - Can use HM Surf outside of battle

Canalave City - Mine Badge - Can use HM Strength outside of battle

Snowpoint City - Glacier Badge - Can use HM Rock Climb outside of battle

Sunyshore City - Beacon Badge - Can use HM Waterfall outside of battle

Hope this helps!

The first gym gives you TM stealth rock
The second gym gives you TM grass knot
The third gym gives you TM drain punch
The fourth gym gives you TM water pulse
The fifth gym gives you TM shadow claw
The sixth gym gives you TM flash cannon
The seventh gym gives you TM avalanche
The eighth gym gives you TM charge beam