You've mentioned verizon and yahoo. How is Thunderbird involved?

I thought Thunderbird helps in the migration of Verizon to Yahoo

If you set up both accounts in Thunderbird, you can simply move or copy messages from one account to another. No "import" as such is needed.

How do I set up both accts. in thunderbird and will I be able to move all the verizon in-box emails, contacts and folders with corresponding emails in them? If this can be done then this is a great I'm sure for lots of people because either the ISPs or their partners like ShuttleCloud can't do all three of the migrations or some of them especially the folders.

To see all of your folders you need to be able to set them up using IMAP. Last time I looked, this was possible with Yahoo, but I have no personal experience of Verizon.

There is no standard format for email message transfer between different programs. Some email client programs can import from specific others, but you haven't told us what you use to work with either Verizon or Yahoo.

Run Thunderbird and it will ask you to set up an account and take you through the required steps.

To set up a second account, in Thunderbird go to File|New|Existing Mail Account…

Moving Contacts between different address books is a different matter. I'd look to see if Verizon support export to CSV ("comma separated values") files. Thunderbird's Address Book can import these.

Is there a reason you need to stop using Verizon?