How to add contact us form in to front-page?
You can use Contact Form Blocks module, in case if you want
to use the core contact form.
Alternatively, you can create your contact form using Webform module which supports embedding the form
in a block, which you can then display wherever you want.
Also, you can display your existing contact form in a custom block,
writing a custom module. In that case, you need to implement 2 hooks:


hook_block_info
hook_block_view


and then use the function drupal_get_form to retrieve the form for
rendering. (See this for some more info)