How did the Prophet's prayers—may God's peace and blessings be upon him—take place, both aloud and silently, and what did he recite in each rak'ah (unit of prayer)?

The manner of the Prophet's (peace and blessings be upon him) congregational (loud) and private (silent) prayers is what Muslims observe today, and it closely resembles what is narrated from the Two Holy Mosques. The Companions accurately transmitted it, and it has reached us through continuous transmission (tawatur).

In brief, the worshipper recites the opening supplication (Du'a al-Istiftah): "O Allah, distance me from my sins as You have distanced the East from the West..." or "Glory be to You, O Allah, and with Your praise...", then Al-Fatiha and another surah after it, and says "Ameen" aloud in congregational prayers and silently in private prayers. Then, he bows, saying "Subhan Rabbi Al-'Adhim" (Glory be to my Lord, the Most Great) three or more times. He rises, saying "Sami' Allahu liman Hamidah, Rabbana wa lakal Hamd" (Allah hears those who praise Him; Our Lord, and to You belongs all praise). He prostrates, saying "Subhan Rabbi Al-A'la" (Glory be to my Lord, the Most High) three or more times, and supplicates. Between the two prostrations, he says "Rabbighfir li warhamni..." (O Lord, forgive me and have mercy on me...).

In the middle and final sittings, he recites the Tashahhud, adding the Salatul-Ibrahimiya (Ibrahimic prayer) in the final sitting. After concluding the prayer with Tasleem (salutations of peace), he recites the prescribed remembrances (adhkar) such as Tasbeeh (Subhan Allah), Tahmeed (Alhamdulillah), and Takbeer (Allahu Akbar) thirty-three times, concluding the hundred with "La ilaha illa Allah wahdahu la sharika lah" (There is no god but Allah alone, He has no partner).