Open your home file, set to show hidden files, find the
.mozilla-thunderbird folder and copy it. When you get the new computer
set up, copy that file back into home. Tis will transfer all of yopu
saved email, contacts, and settings for the mailbox.

Well just click on the TB folder, select one message header or all of
them by holding the shift key down between first and last selection.
Then righ click and Save As to your HD. From there is easy to put them
on a CD or Memory pen

it is safer, so that you know you didn't forget anything, to save
everything under ".mozilla-thunderbird".
This way you should be able to remember what extensions you had also.
Of course the OP and you can save how you want to but if you just want
emails then you want to save everything under the "Mail" folder within
the "********.default"
folder.
For future reference you could setup a gmail account and using imap save
all your important emails to that account and not have to worry about
where they are or if you accidentally delete the local copy.

Long winded way to go about a natural housekeeping task??