Settings<Mail, Contacts, Calendars - select your mail account, and make sure sync switches are enabled. afterwards make sure default account in all sections of the mail, contacts, calendars is gmail.
also make sure you have gmail selected to show in apps. for example if you are syncing google calendars, then open calendars, and select accounts up in the top left corner. verify there is a check mark next to the accounts you want events to show.
Some time agao, Google made a change for non-apps accounts - i.e. those that are free. They no longer work as Exchange accounts so the best way is to set up synching separately for email, calendar and contacts. It's better to do it directly from the device.
Set up cal using CalDav:https://support.google.com/calendar/answer/151674?hl=en
Set up CardDav: https://support.google.com/mail/answer/2753077?hl=en
Set up email: https://support.google.com/a/users/answer/138740?hl=en
I have an account on the iPads 2 , with old events That have synchronised thru gmail. Events posted on IPad and iPhone goes to gmail calendar and are seen on iPhone and samsung phone. But the calendar on the two IPad do not get the information from Google calendar. The two IPad use a different gmail account wich are connected. The iPADS and the iPhone all use 6.1.3 version. My problem seems to be with the iPads, they post but do not read gmail calendar. All the iPads and iPhone and samsung synchronise thru local WIFI, telephone operator not invoved here nor is the WIFI. The password on one of the gmail account has been changed recently.