For newsletters, you will want to begin by creating an account in Mailchimp, Constant Contact, or other 3rd party management application. With many of these services, you can setup a free entry level account.
Mailchimp
For example, Mailchimp will provide subscriber lists, templates and much more for email newsletter management. Once you have your Mailchimp account setup, there are easy list subscription tools in mailchimp that will allow you to create and embed a signup form into your website.