Ensure you have a Google account and can view your calendar via the internet (preferably not on your IPhone). Make a note of your username and password.
Go into the Settings of your Google calendar and ensure you have set to private shares.
On your mobile phone go to http://m.google.com/sync where you should be able to log into your Google account and select or manage your device(s). Select the device you wish to sync and you get to the Settings page. Tick which calendars you wish to sync to your IPhone and ensure your press on the Save button.
Finally, open the default IPhone calendar app and select Calendars. Hopefully, you should now be able to select which calendars you wish to view and relate them to the different colour options available.