Get your Google contacts to sync with your iPhone or iPad. When you set up your Google account on your iPhone your mail and calendar are automatically configured, but contacts are missing.
To manually configure it, go to the Settings app > Mail, Contacts, Calendars > Add Account… > Other > Add CardDAV Account. For the server use “google.com” and then use your Gmail or Google Apps address and password.
Note that this will not keep your iCloud contacts in sync with your Google contacts. If you are using Google contacts you should probably turn off your iCloud contacts to avoid confusion.