Icegram brings a great solution for simple Email Subscriber list building, message management upon new subscriber notifications to all parties involved. It is what I use on this site to manage my email base and send a broadcast message or automated new blog post notifications via eMail!
New subscriber, subscribes:
Subscribed successful message:
Admin receives this email below:
New subscriber receives this email:
Admin receives cron process successful notification:
You have an entire dashboard for managing users, templates, messages, and campaigns.
I hope this helps you setup your own Newsletter system for your WordPress website.