Your question presents a number of interesting issues. I hope you are asking this question out of general concern for your privacy and not for an illegal purpose.
First and foremost, what countermeasures (if any) you would use depend on your security requirements. Your level of paranoia/fear will vary according to whether you are seeking to avoid detection for activities that are merely embarassing (i.e. cheating on your spouse), or mildly criminal (you sell bags of weed to your classmates) or very criminal (you are a member of the Mexican drug cartel). An adversary in a civil lawsuit might be able to subpoena your cell tower records; it would be trivial for a government agency such as the DEA or NSA to get these records if you were of interest to them.
Phones that connect to the cell phone network must connect through a cell site, and the location of the site is recorded and correlated with the serial number of the phone. Many phones can be used without connecting to the cell phone network by using only public wi-fi connections. Telephony apps exist for phones having only wi-fi connections. However, your phone might generate and transmit GPS coordinates (attached to Tweets or Facebook posts) and the phone's location would be somewhat trackable by IP address.
The DEA and NSA are known to be able to detect the use of a newly acquired "burner" phone by correlating metadata among the people you regularly contact. If you are in contact with the same phone numbers as with your old phone, for example your fellow gang members or dope dealers, it is trivial for a law enforcement or intelligence agency to discover your new phone. A CIA official recently gave a presentation in which he stated that phones can be tracked if turned off provided the battery is still attached; and that some phones having kinetic input can be tracked according to the user's gait as she walks down the street!
The key question here is whether it would be worth your while, or that of your associates, to use unusual or "tinfoil hat" methods of attempting to mask your location or the phone's serial number. Unless you are doing something that is certain to attract serious law enforcement interest, and even if you are, you are probably better off not attempting to use methods such as using and repeatedly switching "burner" phones. You might well become more visible than if you did nothing.
Finally, don't forget that your movements can also be tracked through fixed and mobile license-plate recording cameras, and GPS-based vehicle systems such as OnStar. It sucks to be a criminal, or to value one's privacy, these days.
For legal advice consult a licensed attorney.