Whether you've been using an email address with your ISP or with a webmail service, you don't need to cancel it (unless you want to). Just forward it to the same Inbox that you're sending your new address to.
Every time you get an email to an old address, simply respond using the new address (in both the From and Reply To fields). Over time, your recipients will make the adjustment in their address books.
You can also use the auto-responder feature on your old email (along with the forward feature) to automatically let anyone who emails you at the old address know that you have changed to a new one.
Of course, any new contacts you make will get your new address. And once you get a Hover email address based entirely on your own name, you never need to change your personal email address again!