Software

Exporting it on the same PC you would go to File > Export > Mail/Contacts.

This will export it to Outlook on the Same PC from there you can export it to a PST file that you can move to another PC.

On the other PC use the import option to use the exported PST file.

First of all, moving from one PC to another is not an upgrade (your title). The 2nd PC is a different license.

But, ever heard of the Mail and Settings Transfer Wizard? It comes with WinXP. You can limit what it moves. Use it to move your Outlook Express mail from one PC to the next. Run OE once on the new PC to make sure it works, sending and receiving a test. Then install Outlook 2003 on the new PC. During install, it will ask if you want to import your Outlook Express settings and mail.