Use the default drupal search form and alter it as needed using hook_form_FORM_ID_alter
/** Implements hook_form_FORM_ID_alter */
function mymodule_form_search_form_alter (& $form, & $form_state)
{
/* Do your alterations here */
}
Reference the Form API to help learn what changes you need to make.
Rplace mymodule in the code above with the name of your custom module. To output the form, in a template, use the below code:
$form = drupal_get_form('search_form');
print render($form);