Al-Tirmidhi (3451) narrated from Talhah ibn Ubayd-Allaah (may Allaah be pleased with him) that when the Prophet (peace and blessings of Allaah be upon him) saw the new moon, he would say, Allaahumma ahlilhu alayna bi'l-yumni wa'l-eemaan wa'l-salaamah wa'l-islaam. Rabbiy wa rabbuka Allaah (O Allaah, make the new moon rise on us with blessing, faith, safety and Islam. My Lord and Your Lord is Allah). Classed as saheeh by al-Albaani in Saheeh al-Tirmidhi, 2745.
This du'aa' is not just for the new moon of Ramadaan, rather the Muslim should say it when he sees the new moon at the beginning of every month. With regard to saying du'aa' every day, there is no du'aa' that the Muslim should say when starting to fast each day. Rather he should simply have the intention that he is going to fast tomorrow.
The intention is subject to the condition that it be made at night, before the dawn comes, because the Prophet (peace and blessings of Allaah be upon him) said: Whoever does not intend to fast before dawn, there is no fast for him. Narrated by al-Tirmidhi, 730. According to a version narrated by al-Nasaa'i (2334): Whoever does not intend to fast from the night before, there is no fast for him. Classed as hasan by al-Albaani in Saheeh al-Tirmidhi, 573. What is meant is that whoever does not intend to fast and resolve to do so from the night before, his fast is not valid.
The intention (niyyah) is an action of the heart. The Muslim should resolve in his heart that he is going to fast tomorrow. It is not prescribed for him to utter it out loud and say, I intend to fast and other phrases that have been innovated by some people.
And Allaah knows best.