Looking for best (proven) way to port Outlook for Mac folders to new email client (possibly Thunderbird)

I've almost had it with Outlook for Mac recently. At least once every week or two I'm unable to retrieve my emails and just have to wait until they magically appear again. It always works fine on a web browser (I unfortunately have a verizon.net email address so I'm using AOL's email servers), but I don't have access to my folders there. I'm looking to possibly move to a new email client, perhaps Thunderbird (unless anyone has better suggestions), but it appears that copying all of my folders from Outlook to a new client is not as easy at it might seem (or should be anyway). Any thoughts or suggestions or proven methodologies would be greatly appreciated. Thanks in advance.