The first thing that stands out the most to me is that you are not grabbing the element correctly. If you are going to use $('#tagSelector') then you need to add an ID="tagSelector" to your element.
<select id="tagSelector" name="tagSelector" multiple="multiple">
For more info on jQuery Selectors, please see:
jQuery Selectors[
^]
The second thing, because you are allowing multiple items to be selected, you could use a forEach approach like this:
$( document ).ready(function() {
$('#tagSelector').change(function(){
var selectedoptions = $(this).find('option:selected');
if(selectedoptions != undefined) {
selectedoptions.forEach(
alert('Value: ' + $(this).val() )
);
}
});
});
Hope this helps :)