How can I download e-mails w/ Mozilla Thunderbird onto my hard drive & upload to new server? My family's company is moving from an old, washed up server to a new cloud server with a different web hosting company.

I need to download emails from the current web host's server for 14 email boxes to my hard drive, so they can be uploaded to the new web host's server!

I have searched different add-ons for Mozilla Thunderbird, but I am unable to figure out the proper way to download individual e-mail profiles and save them on my hard drive.

I have Googled, but tutorials I have followed are too convoluted or have not worked.

Any guidance in this regard would be greatly appreciated!
The two best tools for shifting email from one server to another would have to be imapsync and offlineimap. Of these offlineimap probably has the best instructions and assistance. 

You could either transfer the emails directly or, with a little setup, download them to your Mac and then back up to the new server.