If I have an Account field and a Contact field under the Cases Module, how can I set it up so the options in the DropDown under the "Contact" field vary depending on the account selected under the "Account" field?

You just need to update the field definition of contact_name in record view this way:

'name' => 'contact_name',
'label' => 'LBL_CONTACT_ID',
'filter_relate' => array (
 'account_id' => 'account_id',
),

Remember to run Quick Repair and Rebuild

Regards