You could add an attribute for “Subscribe to Newsletter” and use an Event hook in your package controller to watch for on_user_add event and check to see if that attribute is checked and if it is do the API calls to sign them up on the 3rd party system.
TLDR: If you are doing the code and don’t have any other custom requirements for the Registration, then the solution suggested by @hutman will be the best and cheapest.
Form Controls and Handlers for custom Login and Registration forms are currently on my ideas list for Form Reform.
The second part, forward form results to a marketing tool would be a custom handler to insert into the form processing pipeline. Forwarding forms to external APIs is a common use and myself and others have already implemented several, so is a well understood development process.
If you have further custom development to do on the registration form, then the Form Reform approach may be better and would be more flexibly configurable in the future.
If I were personally implementing I would follow the Form Reform approach because it extends the toolkit.