Phones are tracking devices. Do not bring your phone, not even turned off because many phones emit Bluetooth beacons and other data that can be recorded and traced.
If you bring a phone, make sure that phone has no idea who you are.
Even if the phone doesn't know who you are, the shop that sold you the phone or the SIM, or the credit card company you paid with, can know who you are. So you'd have to use cash. Even without these, your movements can be tracked through a burner phone and informed guesses made about who you are (e.g. if the phone has been at your home or with your friends).
Turning off your phone doesn't necessarily protect you from tracking either:
Ok I stand partially corrected, it's something that's coming but is not rolled out yet and looks like it'll need the latest phones so everything slightly older is still fine
It is actively rolled out right now all the way back to iPhone 11 (2019) while the device is powered off. Version 16 is current, and the power "off" tracking was backported to older devices.
Android support is spottier. We've had powered off features one OnePlus for some time, such as the ability to trigger alarms while turned off, but more advanced features like location tracking are much more recent to Android because it usually requires specific hardware support to operate while using almost no battery. Apple has the privilege of vertical integration, so they were able to update older firmware.
I think this trend is very concerning, because with no user-servicable battery, we're essentially forced into having our phones on to some degree at all times.
Never underestimate how far they will go to track your movements, habits, etc. It's not even about "the gubment spyin' on me". It's about how valuable that data is to corporate assholes who like to target you with customized advertising, and resell your data, etc. (And yes, as a side-effect, the police can also sometimes take advantage of this ubiquitous data capture).
We live in a time when even our stupid cars spy on us: