You want to use your domain to receive and send emails at Gmail?
Understood. Which e-mail hosting company do you want to do this with? If it's Gmail, then you will need to sign up at google.com/a/ for google apps, and then add your domain there. once done, we will point the mx records for your domain via your dns provider to Gmail.
This video is from 2011, Google Apps did allow people to use their services for free for private domains, but that is not the case any more. You will have to sign up with them, pay them to subscribe and then set it up.
Also this video shows how to add your existing account to Gmail, but since you want to totally route mails through Gmail, you will not be able to add. Nothing will exist on your current server if you change the MX Records. Everything will be via Google.
You can send and receive into your existing Gmail account, but the mails will first go to your current server and then Google will download them. This doesnt involves MX records etc.
First you will need to sign up with Google at google.com/a/ and add your domain there. Once that is added, you then need to login to your hosting or where the DNS is configured, and set the mx records to the following if its Gmail:https://support.google.com/a/answer/174125?hl=en
I can instruct with setting up with a host, or anyone else but i cannot suggest a cheaper solution. I personally use nexcess.net but i do not host my emails with them, i use Google Apps myself, reason being they are secure and my mails do not go to Spam. Most cheaper solution results in mails being filtered as spam.
If you go with Nexcess, they are cheap with hosting but there are no guarantees about spam. You will only use the host to send mails.
Thanks for letting me know.