You can add an onsubmit handler. Without using a third-party library such as jQuery, here's a basic way to do it with inline JavaScript:

<form onsubmit=document.getElementById('first').style.display = 'none';document.getElementById('second').style.display = '';>

The onsubmit is triggered whenever the form is submitted, be it by clicking the Submit button, programmatically, or if a user hits Enter in a textfield, for example.

I would, however, recommend you use jQuery and a more unobtrusive approach than this inline approach though. If you want to see how to do that with jQuery, here's a jsFiddle that shows one way of accomplishing this. Basically, you would add an id such as myform on the form element and then add this to your JavaScript:

$(document).ready(function() {

    $(#myform).submit(function(e) {

        $(#first).hide();

        $(#second).show();

    });

});