A vanity e-mail address might look like joe@smith.com or contact@thealanraul.com.
The basics...
Buy a domainLock it up for as long as possible since prices will increase
Pick a .com extension
I recommend GoDaddy
https://www.godaddy.com/domainsChoose a E-Mail serviceDo not use POP3
I recommend
Microsoft Exchange OnlineThis service will sync your e-mail, contacts, calendar and notes to all your devices regarding of the operating system. You can also log into your account via a web browser. This service is re-sold by many companies including GoDaddy.
https://products.office.com/en-us/exchange/exchange-onlineAnother option is
G Suite by Google Cloudhttps://gsuite.google.com/