Well, with what you are doing, it's actually...
$POST['themonth'] = x (where x is the option selected. Note, you want to use $GET instead of $_POST if you are using the get method.
On your action page, you'd do something like this (assuming you are passing a numeric month value which is recommended):
$date = strtotime($_POST['theyear']."-".$_POST['themonth']."-".$_POST['theday']);
$newdate = $date + (86400 * n); // where n is the number of days to add.
$newdateformatted = date("F d, Y", newdate);
echo $newdateformatted;
If you want to subtract dates instead, the math is basically the same, but you subtract instead... Also, note, this will only work with dates later than January 1, 1970 since it uses the strtotime function. Let me know if any of this doesn't make sense, doesn't work, or you need more guidance.