Hi,
Powering on Xbox 360 controllers is the same as powering on a Xbox One controller - press the guide button until it flashes green. The guide button is silver on Xbox 360 controllers and white on Xbox One/Series X|S controllers.
Support guides for Xbox 360 controllers can all be found on the Xbox Support website (https://support.xbox.com) simply by typing ‘Xbox 360 controller’ into the search box. I’ve added them here though, to refer to.
How to turn on/off your Xbox 360 console and wireless controller
And for wired controllers:
Connect a controller to your Xbox 360
More Xbox 360 controller guides:
Xbox 360 wired and wireless controllers
Set up an Xbox 360 Controller for Windows
Xbox 360 Wireless Controller disconnects or can’t connect
How to replace the batteries in your Xbox 360 Wireless Controller
For help with a Xbox One/Series X|S controller (I understand you’re enquiring about Xbox 360, but I’ve also included the guides for Xbox One in case needed, and to help others):
Connect an Xbox Wireless Controller to your console
Get to know your Xbox One Wireless Controller
Troubleshoot your Xbox Wireless Controller
Update your Xbox Wireless Controller
This applies to both Xbox One/Series X|S and Xbox 360 controllers:
If the controller light is flashing and doesn’t eventually turn to a solid colour, then this means something is preventing it from connecting/powering on (it’s possible it could be all these factors and reasons so be sure to consider this and verify everything):
1. You need to replace the batteries.
2. It’s out of range from your console - try moving closer.
3. It’s not synced to your specific console - try resyncing it, referring to the guides above. The re-sync should be instant but can take a few seconds.
4. The console is off (if the controller is synced to your console, then the power supply would be disconnected, as turning on the controller automatically turns on the console).
Sometimes the light is incorrectly flashing (indicating that it’s resyncing even when it isn’t), although if the controller is working okay, then you can ignore the blinking light - you may want to restart your console if this persists.
Hope this helps! :)