Sending e-mails is an easy task, getting them delivered is another story. How do you properly send e-mails in WordPress with Mailgun?
This is easy, first head over to https://www.mailgun.com/ and create an account there. Mailgun provides 10.000 emails per month for free, after that you will pay a very low price per 1000 e-mails. You will have to configure some DNS records in your domain to activate your mailing service.
Next install this plugin to your WordPress installation: https://wordpress.org/plugins/mailgun/ and head over to Settings -> Mailgun to configure your API key. You can also test your configuration by pressing the Test button.