Authentication for Web application

Hi All,

I want to a create web application in sharepoint 2013. I am very new to Sharepoint.

I am bit confused in Authentication type. Either am I need to use claims authentication or forms authentication.

Anybody help me please.

My requirements are

       We have one site in SP2010.

1. We have nearly 1200 users in AD. Only AD users will use this Website. And they can access this site anywhere from the world.
2. If any user forget their password we will reset in AD. That will Effect the SP2010 site logon.
3.  windows logon(if they are working in the company, their laptop/desktop password also change regardingly)
4. Wise versa ,I mean if they changed their password in the website , that will change in AD,Windows Logon aswell.

For this happening we have logon page with the option I want to change my password after logged on. If clicked on this will take to the password change page their they can enter old,new password and hit change password button.

To obtain all of these am I need to use  claims authentication.

And is it possible to avoid users who ever logged in domain network , skip the logon page. Only if they are outside of the company need to enter credentials to the website.

How to Achieve this .  And For Custom Logon pages any ideas please.

Anybody help me please.

Thank you Very much.