The following is a tip for new iPhone users (and even veteran users too). For Gmail users, there are two ways to setup your email on the iPhone:
Setting up your Gmail account via Google Sync/Exchange is a fantastic option–except by default, only your Inbox is selected to Push emails. If you utilize Labels in Gmail, your labeled messages will not download unless you manually select them. Many think there is an error with Gmail because no labeled messages download.
If you’ve encountered this Google Sync “error”, here is the quick fix:
1. Settings –> Mail –> Select your Google Sync account.
2. Mail Folders to Push along the bottom will say Inbox. Tap it, then proceed to select all the Labels you want Pushed to your account.
After you finish the above steps, selected Labeled emails will begin downloading immediately to your iPhone. Load Mail, go to your Labels and you should see up to date emails there waiting for you. Voila.