I've set up SendGrid in order to better understand issues we are having with our newsletters. In the Settings -Outbound Mail, I am able to Save and Send Email. However, when I go to Mailing > New Mailing, I get: This page isn’t working www.oursitename.org is currently unable to handle this request. HTTP ERROR 500
More often than not, I find 500 errors to be server errors where I need my host's help with it. It could be that a php setting is not high enough, such as memory allocated or execution time on a script. I also get it when I am running into a security issue on the server. Usually tech support has me try the process while they are working with me so they can see the server logs in real time. They just need your IP address first so that they can follow along.
What CMS are you using? If it's Drupal, I can tell you real quick how to find your PHP settings. Not sure if it is in the same place in the other systems.