You can quit google without it affecting you in any way. And you should quit Google.
But since you probably have an Android phone (without an official Firefox OS port) you are surrounded on all sides by Google. But since Google took help from everyone in making Android a success by making it "open source", we have CyanogenMod, Android Open Source Project (AOSP), etc coming to our rescue.
Here is how you get rid of Google from your phone.
Easy way - which does not work
- Trust Google that it won't send your data via backdoors.
- Disable Google search, Gmail, Google Now Launcher, Google+, Google Hangouts, Youtube, Google Play Store, Google Play Music, Sync on all google accounts, everything with G on it via Settings -> Apps -> all apps -> app name -> disable (because uninstall is disabled)
Slightly hard way - easily reversible - incomplete
Root your phone
You most definitely have to find a way to root your phone by yourself. There is no one-click method for all the models. Just go to forum.xda-developers.com and search your phone model and find the root method. They usually have really detailed tutorials.
Install AFWall+ in whitelist mode and allow only your nice apps to communicate with internet.
Harder way - retains android - more complete
- Flash CyanogenMod. xda is where you go again.
- Do not flash gapps (Most xda tutorials will end with a step "flash gapps for google apps"). Our purpose of installing CyanogenMod was to get rid of Google apps. So it makes no natural sense to flash the same.
- Install f-droid and alternate apps like:
- K-9 Mail
Find a FirefoxOS build for your device or port one. Flash that.