You may have noticed that our themes come with multiple methods for customers to sign up for a mailing list on your site, including newsletter popups and footer forms. By default, these forms simply add customers to your Shopify customer database with the "accepts marketing" flag on their account.
Most major mailing list providers, such as Constant Contact, Klaviyo, Omnisend, etc. offer custom apps in the Shopify app store that will automatically sync your customer list to their software without any involvement on your part. Please search for your provider in the app store to see if such an app exists.
In March of 2019, MailChimp announced that it will no longer have a relationship with Shopify and have removed their official app from the app store. If you use MailChimp and wish to continue doing so, there are other options and apps that will still help you synch data between Shopify and MailChimp. Please see this forum post in the Shopify community for more details and alternatives.