[With apologies to Parkie]
Smurfchops:
If you view your email by going to a web page (in Firefox, for example), it doesn't matter which computer you're using. (You could visit a friend in Australia and use their computer. You'd still be logging into the same web page).
If you use an email client (such as Outlook Express, Windows Live Mail or Thunderbird) you'd need to export your emails to a USB stick (or other external media) and then re-import them into the email client on your new computer. [I'd need to know which email client you're using to be able to provide you with specific instructions].
To transfer your Firefox bookmarks from one computer to another, export them from the 'old' computer as an HTML file, which you can store on a USB memory stick. Instructions here:
http://support.mozilla.org/en-US/kb/export-firefox-bookmarks-to-backup-or-transfer
Then plug the memory stick into the 'New'computer, open Firefox and repeat the above procedure but using the 'Import Bookmarks from HTML' option.