What email client do you plan on using?
If you are using Windows Live Mail (it's free) the messages can be imported directly. Simple case of
1) Locate the .dbx files where Outlook Express is saving your mail. Usually here...
C:\Documents and Settings\
username\Local Settings\Application Data\Identities\{RANDOM STRING OF CHARACTERS}
This is a hidden folder. Make sure to adjust your settings so it's visible
2) Copy those files to your new PC
3) Open Windows Live Mail on the new PC. Go to File > Import Messages > Point it to the files and go
That's a very basic guide I know. Google if you need more info there are plenty of guides. Here is a video
http://www.youtube.com/watch?v=vXk_NLO9xoc
For Thunderbird just install it on the same PC as Outlook Express and it can import for you. Then use Mozbackup to transfer it over to new PC - http://mozbackup.jasnapaka.com/
Outlook 2007/2010/2013 is a little more tricky.
PS.
Do you also need to copy contacts or calender?