How do I add a sign up form to my blog?

If you are searching for a simple email subscription without a paid marketing service, then you can use Jetpack subscriptions. A good tutorial is explained in this page

http://www.wpbeginner.com/wp-tutorials/how-to-add-email-subscriptions-for-your-wordpress-blog/

Skip to the Jetpack section, Or you can also use a wordpress plugin for email subscriptions with more control over the newsletters and subscription using

https://wordpress.org/plugins/email-subscribers/