I'm new to the SharePoint 2010 world, and am wondering, is there a log or list somewhere that will show me (as an admin) a log of outbound mail sent by SharePoint?
By default, there is no logging mechanism in SharePoint for outgoing mails.
Mails in SharePoint are sent by giving an SMTP sever IP address in Central Administration. SharePoint never sends email directly. It relays or routes mails to the SMTP server. So, logging should be done in the SMTP server itself.
There is also another way of doing it. You can enable SMTP service of IIS for sending emails from SharePoint, which would also just relay the mails to an SMTP Server. But in this case, you can enable logging in your IIS SMTP Service. Please visit this blog for more details.