Your current function do a lot of unnecessary manipulations: picking month, day and year from the Date object, coverting from number to string and back, splitting, concatenating, updating on several conditions... IMHO, it's too long and unefficient.

So I'd suggest you shorten it using toLocaleDateString method of Date object prototype (see comments in the code):

console.log(
  (new Date()) // take a new Date object.
               // BTW, you can take another time just by 
               // passing any valid time string as a parameter
               // e.g. new Date('7/8/2008')

    .toLocaleDateString('en-US', {year:'2-digit', month:'2-digit', day:'2-digit'})
     // take a locale string from the Date object 
     // passing locale 'en-US' and formatting options as parameters
     // So this returns you a string like "07/08/08"
                             

)

You can do:

function formatDate(date) {
    var d = new Date(date),
        month = '' + (d.getMonth() + 1),
        day = '' + d.getDate(),
        year = d.getFullYear();

    if (month.length < 2) 
        month = '0' + month;
    if (day.length < 2) 
        day = '0' + day;

    return [year, month, day].join('-');
}
 
console.log(formatDate('Sun May 11,2014'));

Usage example:

console.log(formatDate('Sun May 11,2014'));

Output:

2014-05-11

Demo on JSFiddle: http://jsfiddle.net/abdulrauf6182012/2Frm3/

I honestly suggest that you use moment.js. Just download moment.min.js and then use this snippet to get your date in whatever format you want:

Copy<script>
$(document).ready(function() {

     // set an element
     $("#date").val( moment().format('MMM D, YYYY') );

     // set a variable
     var today = moment().format('D MMM, YYYY');

});
</script>

Use following chart for date formats:

To answer part 1 of your question, you current JavaScript probably contains some script similar to this to get the system date: var today = new Date();and something like this to isolate the year:  var yyyy = today.getFullYear();To convert the year to 2 digits, change the value returned for the year above into a string and then create a substring of that using the last 2 digits like this: var yyyy = today.getFullYear().toString().substr(-2);For part 2, I need more info. How are they choosing the date? Do they type in the full date in an entry field or fields? Do they click a date on a calendar? Do they choose a date from a list of available dates?

Some answers don't quite solve the issue. They print the date formatted as mm/dd/yyyy but the question was regarding MM/dd/yyyy. Notice the subtle difference? MM indicates that a leading zero must pad the month if the month is a single digit, thus having it always be a double digit number.

i.e. whereas mm/dd would be 3/31, MM/dd would be 03/31.

I've created a simple function to achieve this. Notice that the same padding is applied not only to the month but also to the day of the month, which in fact makes this MM/DD/yyyy:

function getFormattedDate(date) {
  var year = date.getFullYear();

  var month = (1 + date.getMonth()).toString();
  month = month.length > 1 ? month : '0' + month;

  var day = date.getDate().toString();
  day = day.length > 1 ? day : '0' + day;
  
  return month + '/' + day + '/' + year;
}

---

Update for ES2017 using String.padStart(), supported by all major browsers except IE.

function getFormattedDate(date) {
    let year = date.getFullYear();
    let month = (1 + date.getMonth()).toString().padStart(2, '0');
    let day = date.getDate().toString().padStart(2, '0');
  
    return month + '/' + day + '/' + year;
}