@AMC123,
Are you using a web page to get to your email or a program like Windows Live Mail? If you are using a program that downloads the messages like Windows Live Mail, the messages are already on your computer, so you don't need to do anything. When you set up your new account, just add that account to the program and you are set. If you are using a web page, there are probably options on the page to download the messages in a text format or you could just forward them to the new account.