Gmail is probably the best email service that comes with tons of smart features that is offered for free to everyone. Here is how you can sign up for a free Gmail account.
Mail2Web offers a free online service that allows you to check your POP3 and IMAP based email account using your web browser instead of an email client software.
If you are forced to provide an email address to access a restricted service or webpage and not comfortable in providing your personal email address, you can make use of temporary email services where you can instantly get an email address that expires after a certain period.