How to get the day of week and the month of the year?

Use the standard javascript Date class. No need for arrays. No need for extra libraries.

See https://developer.mozilla.org/en/docs/Web/JavaScript/Reference/Global_Objects/Date/toLocaleDateString

var options = {  weekday: 'long', year: 'numeric', month: 'long', day: 'numeric', hour: '2-digit', minute: '2-digit', second: '2-digit', hour12: false };
var prnDt = 'Printed on ' + new Date().toLocaleTimeString('en-us', options);

console.log(prnDt);

As @L-Ray has already suggested, you can look into moment.js as well

Sample

var today = moment();
var result = {
  day: today.format("dddd"),
  month: today.format("MMM")
}

document.write("<pre>" + JSON.stringify(result,0,4) + "</pre>");

<script src="https://cdnjs.cloudflare.com/ajax/libs/moment.js/2.13.0/moment.min.js"></script>

Yes, you'll need arrays.

var days = ['Sunday','Monday','Tuesday','Wednesday','Thursday','Friday','Saturday'];
var months = ['January','February','March','April','May','June','July','August','September','October','November','December'];

var day = days[ now.getDay() ];
var month = months[ now.getMonth() ];

Or you can use the date.js library.

EDIT:

If you're going to use these frequently, you may want to extend Date.prototype for accessibility.

(function() {
    var days = ['Sunday','Monday','Tuesday','Wednesday','Thursday','Friday','Saturday'];

    var months = ['January','February','March','April','May','June','July','August','September','October','November','December'];

    Date.prototype.getMonthName = function() {
        return months[ this.getMonth() ];
    };
    Date.prototype.getDayName = function() {
        return days[ this.getDay() ];
    };
})();

var now = new Date();

var day = now.getDayName();
var month = now.getMonthName();

One thing you can also do is Extend date object to return Weekday by:

Date.prototype.getWeekDay = function() {
    var weekday = ["Sunday", "Monday", "Tuesday", "Wednesday", "Thursday", "Friday", "Saturday"];
    return weekday[this.getDay()];
}

so, you can only call date.getWeekDay();