Currently if I click on login button in k2 user module without typing in username and password, it will redirect me to a sign_on page, but I don't want to go to default sign on page, I want to direct the site to a customized login page. Where I can change the code to do it.

It will redirect to the default com_users page on a failed login.
If you created a custom com_users page then you should map it to a menu item so redirection will be made automatically.

Can you please be specific? For instance, where I can find this page, I know the procedure to customize the template, but what's the procedure to customize a page, just edit the page itself or I need to follow a procedure..

You customised the login page, right?

If not you need to customise the com_users login page.
www.joomlashine.com/web-development/customize-joomla-extensions-with-override.html