As the admin of my site, I currently get notified when someone submits an entry via safecracker, but is there a way to specify which other emails or users get notified on submission?
The default functionality using the channel preferences is not very informative for users. Clients almost always complained when I used it. I highly suggest you checkout Postmaster. https://objectivehtml.com/postmaster
There quite a few add-ons to add additional email functionality to EE but none come close to the robust feature set of Postmaster. You can use any email service to send any array of emails at any time.
Postmaster supports things like re-occurring emails, custom email messages made up of other EE tags, post-dated emails, and so much more. It's really the definitive solution for using ExpressionEngine to programmatically send email at really anytime you desire.
Other add-ons that have similar functionality are:
But if you have similar clients to mine, you will find the other solutions inadequate in flexibility and extendibility. Postmaster is definitely a premium solution with over 6 months of development - all the people that currently use it, just love it.