Is there a tool that permits a person to manage their retail email listing?

I have a client that says their email is always a mess.

1 - They want a method for sending emails to their mailing list.

2 - They want to send a templated email newsletter to different groups within
their email list.

3 - They want an "Add To Mailing List" feature on their site.

I get confused trying to figure it out because it seems like the list
management would be built into their email client.
BUT - it also seems like I could build in some sort of management tool (an
extension) into their website.

Any suggestions - I'd be thrilled.

Thanks!
Tg