Whatever your reasons, protecting your pricacy and information even from big tech companies is never a bad idea. If you know enough about doing that to avoid using Chrome but aren't so gung ho about it to go the full VPN and TOR route here are some very simple steps you can take to peel the Google out of your Firefox browser. It's not hard.
#1 - Go to tools>options and switch to the Privacy and Security tab.
Here you want to toggle on "Custom" and do not toggle on any of the settings shown under custom. The reason for this is so that you can also set the "do not track" to "only if Firefox is set to block known trackers" in a way that will not contact Google. You don't need Firefox to contact Google to look up any lists or telling any website anything.
You do not need Firefox creating, maintaining or searching for any Google lists in deciding what to block and accept. For this use a 3rd party addon like uBlock Origin(recommended). You don't need Firefox core looking up any Google lists or checking with Google(or anyone) for anything if you use a 3rd party addon to do this job.
Checking a google list, be it for trackers or trusted status, with every url you visit essentially tells Google about every URL you visit. The above stops this(right now, FIrefox changes setting effect regularly)
section #2 - Cookies and Site Data
Check "Delete site cookies and data when Firefox is closed". This will delete all cookies every time you close the browser, no need to share all your cookies with every beacon you run across. For useability purposes click on the "manage permissions" tab and allow the login pages of any site you want to remain logged into permanently. You will stay logged in without keeping non-login cookies.
Never use the keep "logins and passwords" feature. Again, grant a cookie permission to the manage permissions tab to keep you logged in but don't store the passwords on your computer, master passwords are never a good idea. This way you will also protect yourself from keyloggers as you will not need to enter passwords to log in after the first time. NEVER do this with financial or sensitive accounts! You don't want any computer, including yours, to permanently stay logged into any account that has vital or financial information about you. It's fine for forums, tools and non-critical sites but the critical stuff... never permanently store login cookies for them.
section #3 - History
Firefox will: "use custom settings for history". The only tab to check here is the "clear history when firefox closes" BUT in the settings tab you will make sure "Cookies" and "Active logins" are UNchecked to make section #2 work. The rest need to be checked.
section #4 - Address bar
Nothing to check in this section, leave it all unchecked
section #5 - Permissions
These are device permissions like camera, turn off everything you don't use. Block sound and popups as you desire.
section #6 - Firefox data collection
Not needed, enough people leave it on by default that Firefox will get all the info they need without yours. You can safely leave everything here unchecked.
section #7 - Security
Leave everything unchecked here. This is where every url is checked against Google lists of attack sites. You share all the urls you visit so that they can be checked by Google against their list. If you don't visit unknown sites and you use 3rd party blockers like uBlock Origin you get this protection already without telling Google what URLs you visit.
section #8 - certificats
Not critical but you can make them ask you every time and not be too inconvenienced.
Next comes about:config
This can mess up your browser so I'll keep it short. Perform a search for the word "Google" and modify all resulting entries with a blank space, or disable if it's an on/off setting.
Lastly the defaults
It's not enough to set an alternate search engine, you need to delete Google as an option under options>search. Some have said the little favicon of the Google logo is loaded on your browser every time you visit a url which tells Google your every URL visited.
This just covers Firefox settings and options, you probably need to do more to block beacons and such but this will get you started with the options you control. Sadly if you deviate from the above you give Google an implicit permission to record and permanently store you browsing activity because any other setting in one way or another will involve Google.
Fix what you can control, Google doesn't need a full report on your every online activity, nor does anyone else.
Note: The settings to keep you logged in by only granting specific sites permission to store cookies can be a bit tricky. The above is correct but in some cases a site will use a different URL than it's main domain, or a subdomain, or even multiple domains for login verification. To figure out which urls to give permission to you can close the browser(all cookies being set to be cleared) and then login and go look at the cookies now on the computer. Remember that http and https are two different URLs, likewise www and non-www are two different URLs to Firefox.
There are two sections that must be set as above to get logins to stick but trust me, they will not stick if you don't set them up as above. This is one area that Firefox recently modified which is counter-intuitive to set up.
Also, when you do something like log into your gmail get in the habbit of closing the browser entirely both before and after checking out your gmail/adsense/other google product so as to clear all cookies and beacons they load up.