How do I refresh / check new mail in Mail?
Apple Mail's check for mail icon looks like a new mail message icon, and can be found here:



If it can't be found you can add by right-clicking on the toolbar and choosing customize:



You can then select the icon from the list and drag it to the toolbar at the top:



You can also use "Mailbox" -> "Get All New Mail" which has a shortcut ⌘ + ⇧ + N.

You don't need to synchronize. Synchronize does more than a new mail check (see this answer)