The safest thing you can do with a cell phone is not have a cell phone. Having a mobile device means that at least your cell phone provider and more than likely the company that makes your OS know where you are at any given time. If you're going to do something you don't want people to know about, leave your phone at home. It is entirely possible to live without a cell phone, but it's okay to have one. Just make sure you use it responsibly and sparingly, and don't use it when you don't have to.
Your Existing Device
Many of the same web browsing tips that are in the above Web Browsing section are valid for mobile web browsing. Firefox mobile even supports browser extensions, to improve your tracking protection. However, a few other browsers are worth using on mobile. Firefox Klar is a version of Firefox Focus (a self-erasing, privacy-focused browser) with some extra tracking protection. Also, OrFox is a version of the Tor Browser for Android, which is good for all the same reasons as the desktop tor browser above.
Outside of iOS, most systems leave messages encrypted, leaving them especially vulnerable to MiTM attacks. One remedy (but not cure) to this is the Silence app, which can encrypt messages sent between two users of the app.
If you're on an Android device and feeling adventurous, installing LineageOS is a good choice to minimize Google's tapping of your phone. Note that this will restrict you to F-Droid unless you add Google apps back in- but what's the point then, anyway?