I was under the impression that once I install Thunderbird, set up my existing email accounts (IMAP) then all my folders and saved messages would be accessible from that computer.

Only IF those folders and messages exist on your mail server.

The folders show up but not the messages.

Is Thunderbird downloading messages? It should, after adding the account. The status bar should show you what Thundebird is doing. If you have a lot of messages on the server, it will take time for Thunderbird to download all of them.

Whenever I save a message on one computer or phone will it then appear on all my devices whe I access Thunderbird? Than does not seem to be happening.

Only if you're saving them to an IMAP folder on the server.

Thanks for the reply however I still do not understand. All the active message on the incoming board are there across all devices. It's the saved messages that are not in the folders which did show up. i did set up IMAP accounts.