What is the process to register a house

Talk to the Comune - they'll advise you on the requirements  

If you are no longer in 'corso di costruzione' there is the accatastamento to pay, the completion of works, and so on - there is no fixed price, its related to the size of the building, where it is and so on.    You need to get your agibilità and then its all done and dusted. 

Technically, you should request the certificate by contacting the "Ufficio Tecnico! of the Comune within 15 days from finishing the works. Here is some general information http://www.professionisti.it/enciclopedia/voce/1120/Certificato-di-agibilita-fabbricati

It is more usually done by the professionista in charge of direction of works when they close the pratica of the cantiere.  You will be paying him to do this, as it will be included in his fee. 

Thanks to althat replied ...