Zoho Mail is a great mail host that supports free custom domains. Here are the steps to get it all put together.
Gmail is a great email web app. They have great spam filtering, and the UI is snappy. In the olden days, they offered free custom domains if you used the Google Apps for Business. That has all been monetized by a subscription these days. The G Suite will cost you $5-10 a month. This is reasonable, but free is better.