JQuery Datepicker, can't trigger onSelect event manually!

$('.ui-datepicker-current-day').click();

Couldn't you just refactor that to a function of its own, which you reuse? Strictly speaking, a datepicker select is not really what happens on page load. You just want to do exactly the same thing that happens when the datepicker is indeed selected.

function populateList(dateText, inst)
{
    alert('alert test');
    $('#date').val($.datepicker.formatDate("yy-mm-dd",$('#date_calendar').datepicker('getDate')));

    // Ajax for populating days when selected
    $.post("server_requests/show_day.php",
        {
            date: $('#date').val(),
            user_id: $('#user_id').val()
        },
        function(data)
        {
            //return function
            $('#my_day_tasks').html(data.resultTable);
        },
        "json"
    );
}

$(document).ready(function(){
  //create date pickers
  $("#date_calendar").datepicker(
  { 
        changeMonth: true,
        changeYear: true,
        dateFormat: 'yy-mm-dd',
        defaultDate: $.datepicker.parseDate("y-m-d", $('#date').val()),
        onSelect: populateList
  }).disableSelection();

  // i'm not bothering to pass the input params here, because you're not using them anyway
  populateList(); 

});

You could try something like this -

    $(document).ready(function(){
    //when page loads                      
    SetDateTime(null);
    //create date pickers
    $("#date_calendar").datepicker(
  { 
        changeMonth: true,
        changeYear: true,
        dateFormat: 'yy-mm-dd',
        defaultDate: $.datepicker.parseDate("y-m-d", $('#date').val()),
        onSelect: function(dateText, inst)
            {
                SetDateTime(dateText);
            }
    }).disableSelection();

    $("#date_calendar").trigger('onSelect');
});

function SetDateTime(selectedDate)
{
    if(selectedDate == null)
    {
        //get todays date
        var dateNow = new Date();
        //if its a single digit day, add a zero before it
        var sDay = dateNow.getDate().toString();
        if(sDay.length == 1)
        {
            sDay = "0" + sDay;
        }
        selectedDate = dateNow.getFullYear() + "-" + (dateNow.getMonth() + 1) + "-" + sDay;

        //load timetable
        ShowDay(selectedDate);
    }
    else
    {
        var ds = selectedDate.split("-");

        //load timetable
        ShowDay(selectedDate);  
    }
}

function ShowDay(dateToday)
{
         alert('onSelect triggered! Yay!');
         $('#date').val($.datepicker.formatDate("yy-mm-dd", $('#date_calendar').datepicker('getDate')));

        // Ajax for populating days when selected
        $.post(
            "server_requests/show_day.php",
                {
                date: dateToday,
                user_id: $('#user_id').val()
                },
            function(data)
            {
                //return function
                $('#my_day_tasks').html(data.resultTable);
            },
            "json"
        );
}