So, what I did for a similar use case:

-Buy domain from name cheap

-Setup email forwarding to my gmail account with namecheap

-Setup gmail to optionally send as [email protected] (I forget the steps, but I think it's under settings -> accounts)

So although the email comes to/from my custom domain, gmail is the real sender so I've not had any trouble with delivery (that I've noticed, at least)

I'm planning a similar setup. Cloudflare now as email forwarding in beta and then use my personal Gmail account to send mail as by custom domain. Here's a tutorial on setting up Gmail's "Send Mail As": https://www.youtube.com/watch?v=MEheS8gM4Xs

I've had issues with Outlook (or whatever Microsoft calls their mail service) placing my emails in the spam folder, so YMMW. Works fine when the receiver is using Gmail, though.

Oh cool, didn't realize that last step was possible with free gmail.