One of the best practices that you may perform at the tomb of the Prophet Muhammad (s) in Medina is tawassul. Here is a narration from a famous Sunni commentary of the Qur'an by Ibn Kathir:
"Many have stated this tradition. One of them is Abū Mansūr Sabbagh who writes in his book al-Hikayat-ul-mashhurah that, according to ‘Utbi, once he was sitting beside the Prophet’s grave when a bedouin came and he said,
“Peace be on you, O Allah’s Messenger. I have heard that Allah says: ‘(O beloved!) And if they had come to you, when they had wronged their souls, and asked forgiveness of Allah, and the Messenger also had asked forgiveness for them, they (on the basis of this means and intercession) would have surely found Allah the Granter of repentance, extremely Merciful. I have come to you, asking forgiveness for my sins and I make you as my intermediary before my Lord and I have come to you for this purpose.”
Then he recited these poetic verses:
“O, the most exalted among the buried people
who improved the worth of the plains and the hillocks!
May I sacrifice my life for this grave which is made radiant by you, (the Prophet)
the one who is (an embodiment) of mercy and forgiveness.”
Then the bedouin went away and I fell asleep. In my dream I saw the Holy Prophet (Peace Be Upon Him). He said to me: O ‘Utbi, the bedouin is right, go and give him the good news that Allah has forgiven his sins."
Source - Ibn Kathir, Tafsir al-Qur'an al-azim, volume 4, page 140, under the Verse 4:64.
Here is a nice recitation of Du'a Tawassul, that addresses the Prophet Muhammad (s) and his Ahl al-Bayt. It also includes the English translation.