How to backup mail with imap, and how do I read it later?

Most mail clients have the option to download all messages when using IMAP.

You don't say what mail client you're using, but in Thunderbird the relevant settings are located at:

Account Settings > [email protected] > Offline & Disk Space

Then you can choose to have the messages downloaded for offline use.

I use offlineimap to synchronize my email between my local Maildir folders on my linux box, and the IMAP folders on my email provider's server.