If you're already using the Mail application to get your email, and all the mail you want to keep is there, that's a good first step. If your account is of the POP type, it has already been downloaded to your computer, and switching accounts with your provider will not delete it. If your account is IMAP-based, you should make sure that you are downloading copies of all mail - go to Preferences - Accounts, click on the account name, then Advanced, and make sure that "Keep copies of messages" is set to "All messages and their attachments".
When you actually make the switch, and your old account is turned off, you will still see the messages in Mail, but you will no longer be able to connect to the mail server - you'll probably see requests for your password (since your old login will no longer work). What you can do is move email from your Inbox (and Sent Mail, too, if you want to keep it) into local mailboxes (folders) which are independent of any mail account. In the bottom left corner of the Mail window, click the + button, create an "On My Mac" mailbox, and drag mail from your Inbox/Sent folders in there. (You can use cmd-A to select all mail in each folder.)
You'll need to add the new account for your new provider in Mail once that's arranged. Again, Preferences - Accounts, and use the + button to add it. Ask your new provider what the server and login settings should be.
Matt
