What device do I add in order to detect if my project is moved?
There's two basic ways of detecting vibration - one which is quantifiable and one which isn't (much).

The simplest way is a sprint (possibly weighted) inside a metal tube. Vibration causes the spring to move and touch the tube - the resultant circuit is just the same as using a button. Sensitivity can't be controlled (other than maybe "how many times in a given period does the spring touch the tube"), but it's very very simple to use. They are often used in simple movement alarms (for things like bikes, etc).

The second method is to use an accelerometer. These are harder to work with but they do give you the actual amount of vibration at any given time. They are far more sensitive than a simple spring+tube, but since they are an active component they need to be constantly queried (often through I2C or an analog input). That makes them harder to use in a low-power scenario.

Some accelerometers are more intelligent and can be programmed to trigger an interrupt when vibration above a certain level is detected, though again that increases the complexity of working with them.

So in short: the lowest power option is the simple spring+tube sensor - it uses zero quiescent current. The best option is a programmable accelerometer that can trigger an interrupt to wake the main MCU. Which you choose depends on how sensitive you need the device to be...