How to setup a mail server on Ubuntu to send and receive mail?

How to setup VPS server to send and receive emails without using 3rd party services (e.g. gmail, hotmail…).

I have a website hosted on my VPS that is using LAMP (Ubuntu 14.04 and PHP 7.1) I installed Postfix and was successfully able to send emails using PHP, however what I want is to be able to read and reply to emails that users send to my website email (e.g. contact@example.com) so I googled on how to do this and most of the answers I was able to find was used some external SMTP like gmail.

My DNS Records look something like this with the exception of using the proper server IP and proper domain name:

|Type |	Name                | Value                                    | TTL        |
|-----|---------------------|------------------------------------------|------------|
|A    |	@                   | 127.0.0.1                                | 600 seconds|
|CNAME|	*                   | @                                        | 600 seconds|
|CNAME|	email               | email.example.com                        | 1 Hour     |
|CNAME|	ftp                 | @                                        | 1 Hour     |
|CNAME|	www                 | @                                        | 1 Hour     |
|MX   |	@                   | mail.example.com (Priority: 10)          | 1 Hour     |
|SRV  |	_autodiscover._tcp.@| 0 0 443 autodiscover.int.secureserver.net| 1 Hour     |
|NS   |	@                   | ns45.domaincontrol.com                   | 1 Hour     |
|NS   |	@                   | ns46.domaincontrol.com                   | 1 Hour     |

I don’t want to use gmail, I want to host and to be able to send and receive email from my own server, and maybe connect contact@example.com to thunderbird or outlook.

Note that my problem is not if I did setup my server correctly but how to setup my server correctly to accomplish sending and receiving emails without relying on services supplied by gmail and others like it.