I have a Facebook page. I want to share my page’s status to fans timeline through a Facebook app, of course after their permission.
Could you show me a tutorial or guide me to the first step for creating such application?
You might be able to do this via IFTT.com
Go through the Facebook developer guides. Your requirements are pretty simple and this is allowed by Facebook.
To create a Facebook application with the requirement you're asking is actually simple and fast. The first thing to know before starting is that Fb does not host applications. To create one, you must have access to a server where your app will be hosted. This server must also be able to run one of the many languages ??supported by Fb: PHP, ASP, Java, ...
So before you begin creating your application, you must add the Facebook Developer App (http://www.facebook.com/developers/). It will give you all the tools to create your applications. Once it is installed, run the Facebook app Developer to create the profile of your future application. Then go to Set up a new Application menu and follow the step. Good luck!