First, ensure you still have access to your rr.com email login (https://rr-emaillogin.com/) Roadrunner account and enable IMAP or POP-access (check settings like incoming server, port 993 for IMAP) as needed. Then sign in to your new email provider (e.g., Gmail), go to its "Import mail and contacts" or "Accounts & Import" section and enter your rr email credentials so it can fetch your existing messages and contacts over POP/IMAP.
Meanwhile set up automatic forwarding (if your old provider supports it) so that new incoming mail to your Roadrunner address is redirected to your new account. After the transfer is done and you've verified everything is intact—including old folders, attachments and contact list—update your important accounts (banks, subscriptions, friends/family) with your new email address, and optionally keep the old Roadrunner account active for a period of time before closing it.