New development: it would appear that for the moment I am getting instant push Gmail.
The problem is with my IMAP account. I sent emails to my IMAP account using three different email accounts (yahoo, gmail, and hotmail) on the computer. Within a minute of being sent, each one popped up in my IMAP inbox. But not on my phone. Now, almost thirty minutes later, all three popped up at the exact same time on my phone.
Any idea what's causing that? I do know that already have my IMAP configured to work in the Apple Mail client on my computer. Is that slowing things down?