With Mailgun, you’ll need to make a few changes to your DNS records - specifically, you’ll need to add a TXT record and probably a couple CNAMES. The actual record changes you need to make will be given to you in your Mailgun account, and it’s pretty simple to set these up through the Digital Ocean DNS dashboard.

I haven’t used Rackspace before, but I am assuming you’re looking at their managed email service. I scanned through their site and it looks like a pretty good out of the box solution, meaning you’d still need something like Mailgun for application emails. It looks like Rackspace is just simple webmail, but you might want to contact them to see if they offer webhooks or anything that you can call from your app. It would be cool if they did, but I’m not seeing anything on the site about it.

I would recommend using Zoho Mail.

If your using/building wordpress websites, use a WP-SMTP plugin to send mail via your email account at Zoho.

Works flawlessly for me and my clients.