Settings, Accounts/mail, look for a CARDDAV option. All you have to do is put your account info in. That's how I did it for my contacts, worked perfectly.
You can only do Exchange if you have a grandfathered gmail account or paid.
The old CardDAV/CalDAV was wrapped into the Google option. No reason to manually setup CardDAV anymore.
Option 1. The best way is:
Exchange: Mail
Google: Calendar/Contacts
2. Since you probably can't do exchange.
Do Google all. Set Fetch to 1 Hour
Download the GMail app and authenticate it. Notifications Settings: turn off badges, and just have sounds and maybe lockscreen for push. But read in Mail.