I am living by myself and I spend Ramadan by myself. I want to know if I am obliged to give Zakah al-Fitr or my dad should do it. Are female responsible to give the Zakah or only men? In what circumstances?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Giving Zakah al-Fitr is an obligation on every free Muslim who possesses enough food exceeding his or his family's basic need for the duration of one day or night of al-Eid. ِِِِAlso every Muslim has to pay Zakah al-Fitr on behalf of everyone he is obliged to spend on, such as his wife, children and parents, …etc.
Imam al-Bukhari reported from Ibn 'Umar that he said: "The Prophet (Sallallahu Alaihi wa Sallam) enjoined the payment of one S'a (2.40 kg) of dates or one S'a of barley as Zakah al-Fitr on every Muslim, young and old, men and women free and slave. He also ordered that it should be paid before people go to the (feast) prayer" .
Therefore, if the questioner is pubescent and possesses the amount of food that exceeds his and his family's need then he should pay the Zakah al-Fitr for himself and for those he is responsible to spend on.
Know that the rulings of Shari'a are common for men and women, except if there is some peculiar ruling proved by Shari'a. For example, giving Zakah al-Fitr for a wife is an obligation on her husband since he is responsible to spend on his wife even if she is rich. The evidence for this is the aforementioned Hadith of Ibn 'Umar .
If a woman is unmarried and she is able to give Zakah al-Fitr then she should pay for herself if her father is not responsible for her maintenance.
As for al-Zakah a woman is obliged to pay from her money since Zakah is given from the money possessed by the person, and does not have anything to do with the rule concerning responsibility of expenditure. Therefore, if the money of a woman reaches Nisaab (the least amount that is liable to pay Zakah) and a lunar year passes after possessing that amount, then the Zakah becomes due on that woman even if she is married since it is her duty not her husband's.
Allah knows best.