If you still have access to the old computer, you can export your accounts messages and settings. Just open the menu, click on Tools, then Export. Copy the file to your new computer and use the same process to import.

I'm no expert, but Different accounts use different methods for login. Do you know if it is POP, IMAP or Exchange? You might get a pop up window for some providers that requires you to log in in another window or use a special authentication (such as using a code that you receive by text message).