The easiest way I know of is to install an email client like Thunderbird. T-bird is free from Mozilla, makers of Firefox. Setup both accounts as IMAP in T-bird. You can then move mail between accounts using cut/paste or select multiple emails (click first, shift-click last), right click within selected items, select "move to" and select a folder in the other account. That's it.
With most IMAP clients you can also put emails into an "archive folder" and back them up to other media (at least Outlook allows this).
Also, if you are running out of space, use thunderbird or outlook to look for the larger emails, and either download the attachments to other media and delete the original, or move those first to clear space. After moving an email, don't forget to empty the trash for the sending account.