You actually DO want to use MS Exchange...it doesn't mean you're using a Microsoft account or email, it's just the name of the syncing service. It's free, very easy to set up, and once you do that you can sync Contacts, Calendars, and Mail over the air. Read about it here:
Daily Tip: How to setup Gmail as Exchange to push your e-mails [Beginner] | iMore.com
I have all my gmail accounts set up this way and can't imagine ever using anything else.
Here are the setup steps from the linked article:
Launch the Settings app.
Tap Mail, Contacts, Calendars.
Tap Add Account.
Tap Microsoft Exchange.
For Email, enter your full Gmail or Google Account email address (i.e.
name@gmail.com or
name@mygoogleaccountdomain.com).
Leave the Domain field blank. (Really.)
Again, enter your Gmail or Google Account email address as the Username. (Yup, that's twice.)
Enter your Gmail/Google Account password for Password.
Tap Next.
When the Server field appears, enter m.google.com.
Tap Next.
Select what you want to sync (Mail, Calendar, and Contacts).
That's it!