You can set up all the necessary mail account settings on a second (or third etc.) computer.
On all but the main (first) computer on the Advanced Tab of the Account Properties you should check the setting "Leave a copy of messages on server". That way when you go to your main computer the messages will still be available for download.
Also, when sending mails from any but your main computer you should consider copying yourself on the email (the BCC field is best used for this, that way others don't see the fact that your copying it to yourself). Then when you receive mails on you main computer, you will receive a copy of mails that you've sent from the other computer(s).
You will need to bear in mind that you won't see all of your emails on any computer other than the main computer (unless you repeatedly copy over [import] the mail folders - but that's laborious).
The alternative to all this is to use webmail on all but your main computer.