I tihnk it's a timing issue; you can use a delay
(or a setTimeout) and your animation will work.
Code:
$(document).ready(function () {
$("#sectionOneButton").click(function () {
$(".subSection, .subMenu").hide("drop").delay(500);
$("#divSubMenu_1").show("slide", {
direction: "up"
}, function () {
$("#divSubSection_1").show("slide", {
direction: "left"
});
});
});
$("#sectionTwoButton").click(function () {
$(".subSection, .subMenu").hide("drop").delay(500);
$("#divSubMenu_2").show("slide", {
direction: "up"
}, function () {
$("#divSubSection_2").show("slide", {
direction: "left"
});
});
});
});