What sensor should I use to determine if my project is moving?
If it's for use on a bicycle then I'd use a low-power opto detection circuit on the spokes of the wheel. There are several variants of this and of importance is the ability to pulse the LED to conserve power thus, if a spoke passes by at 10 times a second (indicating a minimum speed) then you'll need to power the LED for at least 0.1 seconds and "look" for a beam interruption - this indicates movement. Just an idea.

You might also consider a saddle switch so that if someone is not sat on the saddle and no movement is detected then the LED shuts down completely.