Contacts backup with multiple Google accounts How can one manage backup when you have one device or two devices and 2 or 3 Google accounts? (Work, personal, family)
Firstly, go to the new Contacts preview page by Google on your PC. Sort through the incredible mess that is sure to be there. Log in with all your accounts and then import and export your contacts from one account to another to segregate them into the proper categories (as you said, Work, Personal, Family). I suggest that you keep a single dedicated account for each category, for the smoothest experience. You can mix usage of all your accounts too, but that would just mean unnecessary hassle.

Then, whenever you save a new contact on your phone, make sure you select the appropriate account from the drop down menu at the top of the new contacts page.

Remember, any time you make a mistake in saving a contact, don't try to deal with it on your phone. You will end up creating a bigger mess. Just use Google Contacts on PC and deal with it thoroughly.