If you are using Internet Explorer, click on File|Import and Export. You can export your favorites to a place you choose anywhere on the computer. The export will be an html page. On the new computer the same File|Import and Export function will let you import the html page and you'll have all your favorites back.
Other browsers should have similar Import/Export functions.
In Outlook Express, you need to go to File | Folders and choose "Compact All Folders" before attempting to save messages.
After the "compacting" is finished, close OE, and use Windows Explorer to search (usually from C:\) for "*.dbx" to find the Outlook Express Folder under your username. You may have to use "Advanced Search" and tell it to Search Hidden and System Files, and be sure that Search Subfolders is on. It's recommended that you copy the entire "Outlook Express" folder and paste it somewhere accessible. Copying individual subfolders isn't always reliable.
On the new computer, OE can use File|Import to "import from external backup." Just browse to the "Outlook Express" folder that you copied from the old machine.
Outlook Express includes a File | Export that allows you to "export messages," and you might want to use it as a backup; but simply copying the whole Outlook Express folder and importing from it usually is more reliable (but not always). The files generally aren't enormous, so you may want to try both methods and make two separate backups.
You also will want to move your address book. You can use the same method as for messages, by searching for "*.wab" and copying to the new machine, then using File|Import to bring it back into the new Outlook Express.
Caveat: If the new computer uses Vista, it doesn't have Outlook Express. You will need to use Outlook.
Outlook cannot import an OE address book in .wab form, so on the old machine you need to use File|Export Address Book and save the addy book in ".csv" format, which Outlook can import.