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From the Windows 8 Mail app (or from People or Calendar) you can connect your Google account. Click the Settings charm and then click Accounts to see this set of options, which includes Google. (You can connect multiple Google accounts.)
After entering your Google account name and password, be sure to click the check box under it so that you can sync contacts and calendar items along with mail.
The Windows 8 Calendar app on the left is showing items from the Google calendar. The Mail app, on the right, is snapped along side, showing available folders in that account. Click any folder to see its contents and read new messages. Useing the app bar, you can pin any Gmail folder to the Start screen for quick access.