Below are five short easy to memorize Ramadan prayers that I picked from the book of Adhkaar by Imam an-Nawawi. I made sure that I picked only authentic narrations (grade of Hassan and above). Below you find the Arabic, the transliteration, the translation, and some commentary when necessary.
What to say the first time you see the new crescent of Ramadan (Ramadan Prayers)
اÙÙÙÙÙÙ Ù٠أÙÙÙÙÙÙ٠عÙÙÙÙØ§ Ø¨Ø§ÙØ£Ù ÙÙ ÙØ§ÙØ¥Ù٠اÙÙ ÙØ§ÙسÙÙÙØ§Ù Ø©Ù ÙØ§ÙØ¥Ø³ÙØ§Ù ٠ربÙÙÙ ÙØ±Ø¨ÙÙÙ٠اÙÙÙÙÙÙ
Transliteration: Allahumma ahillahu alayna bil-amni wal-iman was-salaamati wal-islam. Rabbi wa rabbuka Allah
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Translation: Oh Allah, make it a start full of peace and faith, safety and Islam. My lord and your lord is Allah

Commentary: Notice how this supplication paired peace and faith as if they are twins. In fact, in Arabic they sound alike (Amn and Emaan) because one is derived from the other. It also paired peace and Islam (Silm and Islam) because again one is derived from the other. In short, the two most important concepts in our religion (Islam and Eman) are both derived from the words that mean peace in Arabic, Amn and Silm. We ask Allah to make this Ramadan full of peace.
What to say when breaking fast at sunset every day (Ramadan Prayers)
Ø°ÙØ¨ Ø§ÙØ¸Ù Ø£ ÙØ§Ø¨ØªÙت Ø§ÙØ¹Ø±ÙÙ ÙØ«Ø¨Øª Ø§ÙØ£Ø¬Ø± إ٠شاء اÙÙÙ
Transliteration: Dhahaba adh-Dhamaâ wabtallatil-urooq wa thabatal-ajr inshaaâAllah
Translation: Thirst is gone, the veins are wet, and the reward is confirmed by the will of God
Commentary: Remember, the time of breaking the fast is one of the times where duâa is likely to be answered. So you can accompany this Prophetic duâa with any other duâa of you are making.
What to say if someone tries to revile you while youâre fasting (Ramadan Prayers)
Ø¥ÙÙ ØµØ§Ø¦Ù Ù Ø Ø¥Ù٠صائ٠Ù
Transliteration: Innee saaâim, innee saaâim
Translation: Iâm fasting, Iâm fasting
Commentary: Imam Nawawi mentions that this either should be said loud so the aggressor can hear and hopefully stop, or it should be said in silence as a self-reminder. But the first opinion is stronger.
What to say after breaking fast with a group of people (Ramadan Prayers)
Ø£ÙØ·Ø± Ø¹ÙØ¯ÙÙ Ø§ÙØµØ§Ø¦Ù ÙÙ Ø ÙØ£Ù٠طعا٠ÙÙ Ø§ÙØ£Ø¨Ø±Ø§Ø± Ø ÙØµÙت عÙÙÙ٠اÙÙ ÙØ§Ø¦ÙØ©
Transliteration: aftara indakum as-saaâimoon, wa akala taâaamakum al-abraar, wa sallat alaikum al-malaaâikah
READ MORE: How Islam Defines True Happiness And Peace at heart
Translation: May the fasting people breakfast at your place, and may the pious eat from your food, and may the angels pray for you
What to say if you encounter the Night of al-Qadr (Ramadan Prayers)
اÙÙÙÙÙÙ Ù٠إÙÙÙ٠عÙÙÙÙÙ ØªÙØØ¨ÙÙ Ø§ÙØ¹ÙÙÙ ÙØ§Ø¹ÙÙ٠عÙÙÙÙ
Transliteration: Allahumma innaka afuwwun tuhibbu al-afwa faâfu anni
Translation: Oh Allah you are forgiving, and you love forgiveness, so forgive me