Things to consider:
1) Any tracking, tracing, or retrospective investigation of your cellphone use will make use of not only the telephone number (which is linked to the SIM card) but the IMEI number (which is linked to the device). Therefore, don't make the mistake of getting multiple SIM cards and using them in the same phone in order to obfuscate your activity, because the call traffic data will all be linked by the same IMEI and so the changing telephone number will be pointless.
2) Any app that uses data (as opposed to voice) as a means of communicating (eg. Whatsapp, Viber, etc) is much harder for the authorities to work with. However, the downsides are that you require a smartphone (ie. more expensive) to use them, and also you do not know the developers of the app, and how they might be storing data without your knowledge.
3) Many public wi-fi hotspots retain MAC address records, so although you might not provide any identifying information when using them, it will be possible for the authorities to link uses at multiple locations and multiple times by way of the common MAC address. This may or may not bother you, I don't know.
4) Be aware that moving around the country with your burner phone switched on, at the same time as having your own, non-burner phone switched on, is a weak point. You might be inadvertently leaving a trace whereby they see that both phones are following a common route, and are therefore in the hands of the same person.
If it were me, taking top precautions, I'd go out and buy a used cheap cellphone and SIM card from a second-hand shop. Leave your own phone at home and travel a couple of miles each time you use the burner, to avoid the link to the same cell tower as your own phone. Remove the battery from your burner whenever it's not in use. When you're done with it, remove SIM and destroy SIM and phone separately. Encryption is great if you suspect your communication might be intercepted, but if you're using a burner from a random source, for short period of time, there's no reason to think that comms from it are being intercepted. (Consider, though, whether the person you are communicating with might be having THEIR comms intercepted.)