I have a shiny and new iPhone 4 and a Gmail account. I’ve set my Gmail accounts up on the phone, but I’m a little confused about the “push” feature. As it is now, I only know if I have new mail if I select the mail icon and wait for it to update. Shouldn’t the push feature work somewhat similarly to text messaging? I was expecting my email account would automatically refresh and alert me if there were new messages. Am I wrong?
Mine does, but I have a BlackBerry; THE KING OF PUSH 
Seriously, maybe it’s a setting.
visit Browse All of Google's Products & Services - Google from your computer for set up instructions. If you’re already using Google Sync, you can just enable push mail.
Once you’re set up, new messages are normally pushed to your phone within seconds. While this type of speed is pretty awesome, push connections tend to use more power than fetching at intervals, so don’t be surprised if your battery life isn’t quite what it used to be. We’ve done a lot of work to optimize power usage, but if you prefer to save battery life, you can always turn off push in your phone’s settings and fetch mail every 30 or 60 minutes instead.
Push notifications can be enabled for various apps. It allows the app to receive information even if you don’t have it running. This page will tell you more about it: Use notifications on your iPhone or iPad - Apple Support
You need to play with your iPhone a little bit more, especially in Settings. Go to “Mail, Contacts, Calendars” and then “Fetch New Data”. Turn it on, and select how often you want it. I have mine on “Manually” because it burns through too much battery checking to see if I have mail.
I finally figured out what was wrong - when I set up my gmail account, I tapped “Gmail” instead of “Microsoft Exchange.” Once I set it up again under Microsoft Exchange, I was able to get it to work.
Thanks everyone for your help!!