Google services like Gmail, Google Calendar, and Google Contacts are incredibly popular and—what's more—incredibly easy to set up on your iPhone or iPad. If you have a free Gmail account, you can set it up through the Settings app and access it through the built-in Mail, Calendar, Contacts, Reminders, and Notes apps. If you prefer Google-brand apps for your calendaring, email, and other services, you can also download their versions and set up your accounts there.
And if you have a paid Google account, you can use either of those options or you can set up your account with Google Sync, Google's implementation of Microsoft's ActiveSync service. (Sadly, Google discontinued ActiveSync support for free accounts in 2013.)
How to set up Gmail, Google Calendar, and Google Contacts via the Settings app
If you want to add your Google account to your iPhone or iPad's system apps, you'll want to go to Settings > Mail, Calendar, Contacts. Under the hood, Apple is using the open standard IMAP, CalDAV, and CardDAV services to handle syncing Google services; unfortunately, Gmail retro-fitted support for IMAP, and it can be quirky at times—especially if you like Gmail's labeling system. If you want to use the built-in iOS Mail app and its unified inbox, however, and you don't have a paid account, it's your only option.
Launch the Settings app from your Home screen.
Tap Mail, Contacts, Calendars.
Tap Add Account.
Google's authenticator screen will pop open, where you can enter your email address and password. You also may be prompted for your two-factor authentication code, if you're using one. (And you should be. Here's how to set one up.)
Tap Allow when prompted with the "iOS would like to..." screen.
Choose which services you'd like to sync: Mail, Contacts, Calendars, or Notes. You probably want most (if not all) of them on.
After you save your account, you can access your settings at any time by tapping the Gmail entry under the Accounts header. If you'd like to change the description from "Gmail" to something more apt, tap the Account button, then tap the Description field.
How to set up the Google Sync version of Gmail, Google Calendar, and Google Contacts
If you have a paid Google account, you can take advantage of Google Sync, the company's version of Microsoft's ActiveSync. You won't get Notes sync when you use Exchange, but you do get incredibly fast email.