Where are the emails? If you've been using web-based mail, you need to set up a POP3 account with your computer's email program and then download all of your messages with that. If you've already been using POP3, the messages won't be lost when you change ISPs. The problem, if you've been using web-based mail, is that retrieving with POP3 won't get your *sent* messages, only the incoming ones.
If you need both received and sent messages, this could probably be achieved using IMAP instead of POP3, but you need to be careful with this or you could lose the lot. With IMAP, messages remain on the ISP's server and your email program "synchronises" with the server.