I'm not sure if you've tried this, but here is where you set your notifications:
1) Open Gmail
2) Press the menu button. It's the left-most button along the bottom with 4 squares
3) Select More>Settings
4) Select your Gmail account you want notifications for.
Note. I have 3 Gmail accounts on my phone, so I'm asked to select an account. If you only have one Gmail account, it might bypass this screen.
5) Scroll to the bottom and under "Notification settings" check the box next to "Email notifications."
If you have that checked already, you might have auto-sync turned off.
1) From any home screen press Menu>Settings>Accounts
2) Select your gmail account
3) Make sure "Sync Gmail" is checked
While you're there, you might want to uncheck any other items you don't use like Drive or Google Play Books.
One final possibility is that you have Data Saver turned on or Background data disabled. I don't know if this would affect your gmail notifications, but if all else fails, give it a shot.
1) From any home screen press Menu>Settings>Battery & data manager
For Data Saver:
2) click Data Saver and make sure "Data Saver" us unchecked
3) Go back one menu and select "Battery mode."
4) Check that you have it preset to "Performance mode".
For Background data
5) click Data delivery from the Battery & data manager screen and make sure "Background data," "Data roaming" and "Data enabled" are checked.
Hope that helps!