There's two basic ways to do what you want :
1: Specify the second animation in a "complete" callback :
$('.contact_block').click(function() {
$('.hide_me').slideUp(400, function() {
$('.contact_me').slideDown(400);
});
});
Demo
2: Specify the second animation in a "done" handler chained to a promise generated by the the first animation :
$('.contact_block').click(function() {
$('.hide_me').slideUp(400).promise().done(function() {
$('.contact_me').slideDown(400);
});
});
Demo
In both cases you will see there's an inner function within an outer function, which will be confusing if you've not previously encountered such. This is something you have to get used to in javascript, where functions are first-class objects that can be defined and passed around like other variables.