Firstly: Sunnah Acts Related to Walking Between Safa and Marwah
1. Climbing Safa and Marwah, and invoking and supplicating on them and between them
When close to Safa, one recites the Word of Allah: “Truly Safa and Marwah are among the waymarks of Allah” (Al-Baqarah: 158).  Ibn `Uthaymin says: “It is possible that he (i.e. the Prophet ﷺ) recited the entire verse. The Salaf would refer to an entire verse with a portion of it. It is also possible that he only recited the portion mentioned, which is relevant, for it mentions that Safa and Marwah are amongst the waymarks of Allah, and that it is Safa that one begins with, and that is what is specifically mentioned. This is because the default is that the Companions (may Allah be pleased with them) transmitted everything they heard. Thus, if they did not mention, “then he completed the verse”, or “then he finished the verse”, then it only applies to the portion mentioned.” (Majmu` Fatawa wa Rasa’il Ibn `Uthaymin, 24/466) One also says: “I begin with what Allah has begun”, only saying this the first time on Safa. One climbs Safa until one sees  Seeing the Ka`bah nowadays is difficult, so it is sufficient to face it. the Ka`bah and faces it. One says Allahu Akbar three times and praises Him: “Allah is greater, Allah is greater, Allah is greater.” Then one says: “There is no god but Allah, alone. He has no partner. His is the domain, and His is praise. He gives life and causes death. And He is Powerful over all things. There is no god but Allah, alone. He has no partner. He fulfilled His promise, gave victory to His slave, and defeated all of the clans alone.” Then one supplicates with whatever one can, raising one’s hands. One repeats this three times. One does on Marwah the same as what was done on Safa. One repeats this in all seven rounds with the exception of reciting the verse and saying, “I begin with what Allah has begun.”
One supplicates and invocates plenty while walking between Safa and Marwah. Amongst such words are: “Lord, forgive and have mercy. You are the Most Mighty, the Most Generous.”
2. Running briskly between the two green posts
It is sunnah to walk between Safa and Marwah except for the span between the two green posts  These are green lights on the sides of the walkway. They are referred to in books of fiqh as two green posts. when it is sunnah for men to run briskly between them. This is true in all seven rounds. Consensus on this being mustahabb was related by Ibn `Abd al-Barr and al-Nawawi.
Secondly: Riding Between Safa and Marwah
1. Riding between Safa and Marwah when one has an excuse
It is permissible to ride between Safa and Marwah when one has an excuse. Consensus on this was related by Ibn `Abd al-Barr and Ibn al-Qayyim.
2. Riding between Safa and Marwah without an excuse
Scholars are of two positions regarding the permissibility of riding between Safa and Marwah without an excuse:
The first position: It is permissible to ride between Safa and Marwah without an excuse, and nothing is due for doing so. This is the position of the Shafi`is and a group of the Salaf. This is the chosen position of Ibn Hazm, Ibn Qudamah, al-Shinqiti, and Ibn Baz.
The second position: It is not permissible to ride between Safa and Marwah without an excuse. This is the position of the majority: the Hanafis,  According to them, one must repeat walking between Safa and Marwah if one rides. Otherwise, one must expiate with blood. Malikis,  According to them, if one abandons walking and rides instead, one must expiate with blood. and Hanbalis.  According to them, if one rides without an excuse, it does not count.