The Ottoman sultans did have a set of traditions, that together formed their coronation proces. This proces is called the happening of "Cülus", the word is the arabic word for "to mount or to sit on" the throne. This tradition were mainly made more systematic under Mehmet the II., when the capital moved to İstanbul. Put some things changed through the years and of course it was not exactly the same every year.
I will to summarize the main event chain, and most central events of the coranation. It were performed at the Topkapı palace and various other locations in the city of İstanbul. This proces started with weapons master "Silahtar ağası" and the hareem master "dârüssaâde ağası" would lead the Crownprince "Şehzade" through to the empty throne room, in the throne room the Crownprince would pray to rakats of namaz prayer, beside the relic of "Hırka-i Saâdet". This relic was believed to be a linen coat weared by the prophet Mohammed, and was brought to İstanbul by Selim the I. In the 16th century after the conquest of Egypt, together with other holy Islamic relics. Meanwhile a throne would have been setup at the Bâbüssaâde gate, the sultan to be would come out in some specific robes and take his place on the throne. After this officials and nobels would in some order congratulate the new sultan, and give a kis to lower parts of his clothes. In the end of this part of coronation the Şeyhülislam (the head Imam of the Ulema(Clergy)) would make finishing prayer. After this the Sultan would go back to his private chamber the Has Oda, after a little rest he would accompanied by the officials and nobels he would participate in the tradition funeral of his predecessor.
After the first part the coronation the new Sultan had to recieve the sword of Osman (this was not relic but symbolic ceremonial sword). Osman the I. Is believed to be the first one to receive ceromy like this, but of course not in the geographic area. This had to be done in the weeks after the fist part of the coronation. The Sultan were obligated to go from the Topkapı Palace to the Eyüp Sultan shrine. On his way he would accompanied by various armed forces, his personal guard, state officials and nobels. This symbolic march would be filled with different types of festivities, ceromonies and so on. After the arrival in the shrine there would be short prayer and Şeyhülislam would put the sword on the Sultan. After this ceromy the new Sultan would visit shrines and tombs of earlier his predecessors and pay his respects. The sword giving ceromoni was called the Kılıç Alayı, and it symbolized the Sultans role as a mighty worrior. Under the ceremonies in the way to and from Eyüp Sultan Shrine the Sultan and his officials would gibe out money and goods to the people. With every coronation officials and armed forces would also recieve a special bonus/tip called the "Cülus Bahşiş". This tipping of the were a tradition started with Beyazid the I. In 1389, and become a important political factor in the coronation. And thus the Ottoman Empire would have a new Sultan. A important detail is that until the 20th century outsiders and non-muslims were forbidden to participate i some of the events. But in some of latest coronations religious leaders and ambassadors also participated in these events. The Ottoman sultans usually received the following titles The Sultan of the Ottoman Empire, Custodian of the Two Holy Mosques and The Qaliph of Islam.
It is also important to know that the Crownprinces would have to travel to İstanbul, right after receiving the news of their deceased predecessor. The princes usually was send to provinces where could learn management and leadership skills after they reach the right age. Manisa were usually the most popular province for Crownprinces and they would spend their years there to prepare for the task of Sultan.
Sources and further reading:
Alikılıç, Dündar. Imparatorluk seremonisi: Osmanlı'da devlet protokolü ve törenler. Vol. 2. Tarih Düşünce Kitapları, 2004.
Brookes, Douglas S. "Of Swords and Tombs: Symbolism in the Ottoman Accession Ritual." Turkish Studies Association Bulletin 17.2 (1993): 1-22.
Eralp, T. Nejat. "Eyüp Sultan and the Girding of the Sword Ceremonies of Ottoman Sovereigns (Taklid-i Seyf)." Eyüp Sultan Symposia I-VIII: Selected Articles.p
Karateke, Hakan T. An Ottoman protocol register: containing ceremonies from 1736 to 1808, BEO Sadaret defterleri 350 in the Prime Ministry Ottoman State Archives, Istanbul. Vol. 2. Ottoman Bank Archive and Research Centre, 2007.