You can quit google without it affecting you in any way. And you should quit Google.

But since you probably have an Android phone (and efforts like Firefox OS no longer active) you are surrounded on all sides by Google.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 LineageOS (previously 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

Slightly hard way - easily reversible - incomplete

Harder way - retains android - more complete