Mailgun is an easy-to-use and very cost effective transactional email provider with great analytics tools. You can get started for free without having to sign up for a subscription.
Follow the Mailgun setup instructions to set up and verify your domain.
You will also need access to your domain control panel to edit the DNS settings. It can take up to 48 hours for your domain to be verified but usually only takes a few minutes.
Log into your Mailgun dashboard and copy your private API key from the sidebar. You'll need this in the next step.