What can I say to my father to allow me to go to the mosque, knowing that he previously allowed me when I was at my grandmother's house? And is it permissible for me to go to the mosque for Tarawih prayers at night or during the day with my sister, knowing that the mosque is only 10 minutes away?

A woman is permitted to go to the mosque for Tarawih prayers and other prayers, whether by night or day, provided she adheres to the Sharia guidelines such as covering herself and not wearing perfume. A father should not prevent his daughter from going to the mosque unless there is a fear of temptation, due to the Prophet's (peace and blessings of Allah be upon him) prohibition against preventing women from mosques, even though a woman's prayer in her home is better for her and brings greater reward.