Praise be to Allah.
Du`a for sighting new moon
Al-Tirmidhi (3451) narrated from Talhah ibn ‘Ubayd-Allah (may Allah be pleased with him) that when the Prophet (peace and blessings of Allah be upon him) saw the new moon, he would say, “Allahumma ahillahu ‘alayna bi’l-yumni wa’l-iman wa’l-salamah wa’l-islam. Rabbiy wa rabbuka Allah (O Allah, make the new moon rise on us with blessing, faith, safety and Islam. My Lord and Your Lord is Allah).” (Classed as sahih by al-Albani in Sahih al-Tirmidhi, 2745)
This du`a is not just for the new moon of Ramadan , rather the Muslim should say it when he sees the new moon at the beginning of every month.
Is there any du`a when starting to fast every day?
The intention is subject to the condition that it be made at night, before the dawn comes. The Prophet (peace and blessings of Allah 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-Nasai (2334): “Whoever does not intend to fast from the night before, there is no fast for him.” (Classed as hasan by al-Albani in Sahih 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 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 Allah knows best.