You can use different email clients on different computers. If you've got the apps set up to use IMAP for incoming mail then you'll automatically see all mail in both apps. However if you've Outlook Express set up to use the older POP3 standard for incoming mail you'll need to ensure that you enable 'leave a copy of mail on server' in order that both clients can see all incoming messages.
To get your contacts from Outlook Express to Thunderbird, first export them to a CSV file:
https://knowledgebase.constantcontact.com/articles/KnowledgeBase/5702-export-contacts-from-microsoft-outlook?lang=en_US#Express
Copy the CSV file onto a USB memory stick (or simply email it to yourself), so that you can have it available for use on your new computer.
Then follow the instructions in 'Step 2' here to import the contents of the CSV file into Thunderbird:
http://www.stellaroutlooktools.com/blog/import-outlook-contacts-to-thunderbird/
(It's not as hard as it sounds, honestly!)