The remember me checkbox on the sign-in form, will allow the studio customer, if she does not log out, to browse my account section and book classes without re-authenticating/logging in. If the customer, upon return, tries to purchase a series, she will be prompted to login.
On the iPhone, customers can use the Safari autofill feature. Here is a link to an article that can help .... http://www.imore.com/daily-tip-setup-mobile-safari-autofill-fill-web-forms