To sync your Google contacts, one good method is using CardDAV. You’ll be able to edit, add, and remove contacts from your device and keep them in sync with your Google Gmail account.
It is straightforward:
Open the Settings application on your device.
Select Mail, Contacts, Calendars.
Select Add Account…
Select Add CardDAV Account
Fill out your account information in the following fields:
-Server: Enter “google.com”
-User Name: Enter your full Google Account or Google Apps email address.
-Password: Your Google Account or Google Apps password. (If you’ve enabled 2 Step verification, you’ll need to generate and enter an application specific password.)
-Description: Enter a description of the account (e.g. Personal Contacts).
Select Next at the top of your screen.
Make sure that the “Contacts” option is turned to ON.
When you’ve finished the setup, the sync will automatically begin.
More about sync your Google accounts:
Bussiness users may need to use the Google Sync app: http://support.google.com/a/users/bin/answer.py?hl=en&answer=138740