Suppose that I domain name and website hosted with GoDaddy and I want to transfer the domain name and website to another host.

The transfer was completed successful. And now I'm using email on my new host.

But my old emails are still on GoDaddy's server. How can I still retrieve my old emails from GoDaddy's server. When I log into GoDaddy as normal, I no longer see my email account there.

Well... assuming that your old account at GoDaddy still exists (i.e. if they didn't delete your mailbox) you should be able to do this using the IP address of their mail server.

Before you moved the domain, you could've found this IP out easily... now I guess the easiest way might be to ask them for it? Once you have the IP address, use that in your email client as the incoming mailserver (e.g. instead of mail.domain.com) and just supply your old GoDaddy mailbox username/password.

Some hosts have a centralised mail system which means you could use something like mail.godaddy.com - but since I don't know which package you have with them, or how they handle their mail for shared hosting (for example) I can't be sure about this part; perhaps someone else could help more on this bit if you gave more details about the package that you have/had with them.

Hello!
Was webmail interface was used? Or you have email client?