I'm assuming you access your mail via https://webmail.earthlink.com' and or https://webmail.peoplepc.com/ and you want to also access your mail with the Opera Mail App/client/program.
If so, in the New Account Wizard in Opera Mail, choose "Email" and click Next.
Then, enter your name and email address and click Next.
Then, enter your username and password, select "IMAP" and click Next.
Then, set the incoming server to imap.earthlink.net or imap.peoplepc.com and check "Use Secure Connection".
Then, set the outgoing server to smtpauth.earthlink.net or smtpauth.peoplepc.com and check "Use Secure Connection" and click next.
Then, the account should be set up. However, goto "Menu -> Mail and chat accounts -> select the account -> edit -> servers tab``` and make sure the outgoing port is 587 and not 25. Then, adjust other account settings as needed.
Now, if you need to transfer your messages to another mail provider (Gmail for example), create another IMAP account in Opera Mail that connects to the other provider. Then, you can drag messages from the IMAP Inbox folder of your Earthlink account to the IMAP Inbox folder of the other account.