How to get the selected date from jquery datepicker

This should do the trick

$(function() {
    $("#datepicker").datepicker();
    $("#datepicker").on("change",function(){
        var selected = $(this).val();
        alert(selected);
    });
});

It's basic but here is a jsfiddle with it alerting the selected date when selected

update to change the date format

$(function() {
    $( "#datepicker" ).datepicker({ dateFormat: "yy-mm-dd" });
    $("#datepicker").on("change",function(){
        var selected = $(this).val();
        alert(selected);
    });
});

jsfiddle

3rd update

$(function() {
    $("#datepicker").datepicker({ 
        dateFormat: "yy-mm-dd", 
        onSelect: function(){
            var selected = $(this).val();
            alert(selected);
        }
    });
});

I have used a little more of the native markup for datepicker ui here try this and see if you get the alert as you are after.

Though, question is answered, for people who just want a date object or set a date with specific format. There is simple functions jQuery provides. Here's working jsfiddle

$( "#datepicker" ).datepicker({ dateFormat: "dd-mm-yy" });

$("#datepicker").datepicker('setDate', '10-03-2020');
                 // pass string of your format or Date() object

$("#datepicker").datepicker('getDate');
                 // returns Date() object

$("#another_datepicker").datepicker('setDate', $("#datepicker").datepicker('getDate'));
                 // pass string of your format or Date() object