To change the email address associated with your account, first go to your settings page by clicking on 'Settings' in the bottom left of any Chess.com page:

Then, click on 'Change' next to your email, or, simply click this link while you are logged in:

https://www.chess.com/settings/change-email

You will then be asked to enter your password, and your new email:

When you submit the form, an activation email will be sent to your new email address. To complete the process, you must use the link in that email to activate your new address.