How to setup Gmail, Google Calendar, Google Contacts on your iPhone or iPad 

The iOS Settings app, Mail, Calendar, and Contacts section contains built-in support for Gmail, Google Calendar, Google Contacts, and related services. Under the hood, Apple is using the open standard IMAP, CalDAV, and CardDAV services to handle the syncing. Unfortunately, Gmail retro-fitted support for IMAP and it's quirky at times. However, if you want to use the built-in iOS Mail app and its unified inbox, and don't have a paid account, it's your only option.

1. Launch Settings from your Home screen.
2. Tap Mail, Contacts, Calendars.
3. Tap Add Account...
4. Tap Gmail.
5. Enter your name, Gmail email address, password, and a descriptive name (like Gmail, or Personal, so you can identify it later).
6. Select which services you want to leave on for sync, Mail, Calendars, and Notes. (You probably want all of them on.)
7. Tap Save.