moving mail to new computer

you can just open the mail app and enter the account's credentials to set it up. everything that is on the e-mail's server (mailboxes etc) will be carried over to the new Mac. be aware that depending on how many e-mails that exist on the server it may take a while to populate them on the mail app.

Migration Assistant will transfer all your Apple Mail accounts over to your new MacBook. I'd start with that.

Transfer to a new Mac with Migration Assistant - Apple Support

Amy mail that is "on my Mac" can get exported as a file and then imported into the new Mac email application.