You want to upgrade from Outlook to Thunderbird and you've changed from Windows to Linux as a desktop. If you have control over a mail server running dovecot, you are in luck.
The tool that you need is the very handy libpst library, although if you run Kubuntu or Ubuntu then this is available on synaptic.
Simply export your Outlook mail into a pst file. Let's call the file mailBackup.pst. Now copy the file to your email server or linux desktop.
It is important to create a directory to work in because for each folder in the PST (Outlook) file, there will be a mbox file created with the same name. Cd into this directory and keep emailBackup.pst in the directory above it.
Now simply run:
The Outlook file will be converted into a series of mbox files. If you run Dovecot, these can be put into your mail directory (be careful not to overwrite existing files!) and you will be able to access them with Thunderbird or your webmail software (I recommend horde).
You are also able to open mbox files in Thunderbird.