It has been a few years since I was really deep in the Crusades research, but for the most part it varied. When Jerusalem was sacked by the first Crusade, all of the non-Christians were killed, including those seeking shelter in religious sites. This led to even more hostilities initially, but many of the zealots soon left to go back to Europe, leaving a small army and a relatively large amount of land to hold. Those responsible for the Crusader/Latin Kingdom, specifically Baldwin III and IV, decided to use diplomacy where possible and allowed Muslims free access to some of their holy sites.
There were, however, those who dismissed this tactic and began to openly assault caravans and worshipers in the name of ridding the holy land of Muslims once & for all. Most famous of these were Guy de Lusignan and Raynald de Chatillon. When Baldwin IV died without an heir, the crown fell to Guy, who was pressured to increase his holdings and deter the spread of Islam through the kingdom. It didn't turn out well and within a few years Jerusalem had fallen back to Salah ad-Din's (also called Saladin) forces.
As to the specific treatment of different sects of Islam, I have no record of them bothering to differentiate them. I do know they were aware of different factions (geographical) but don't recall anything regarding religion branch. May have some luck in the writings of Baha ad-Din ibn Shaddad, Salah al-Din's historian. His work, though slanted at times, is one of the best regarding the Islamic world as it related to the Crusades.
*My apologies for the wiki articles - it's merely meant as a quick reference.
To be clear, the Kingdom of Jerusalem was always at war. The rules which govern jihad do not allow peace between Muslims and non-believers, though they do make exception for a 10 year truce. Various muslim commentators have interpreted this differently, but it seems to have been more or less accepted through the crusading period.
That said, for the first fifty or so years Jerusalem had a relatively peaceful relationship with its immediate Muslim neighbors, though this had little to do with religion and more to do with local politics. Specifically, in the mid-12th century Jerusalem had a tactical alliance to resist the territorial ambitions of Nur ad-Dīn, Ṣalāḥ ad-Dīn's (Saladin) grandfather. This alliance, and the hope for any other such alliance between Christian and Muslim, was effectively shattered by the arrival of the remnants Second Crusade and its decision to lay siege to Damascus in 1148, which should probably go down in history as one of the dumbest military endeavors of all time.
Besides the initial slaughter which occurred when the crusaders took Jerusalem in 1098, the Christians basically took the established procedure for inter-religious relationships as their own. Non-Christians were taxed, prohibited from evangelizing, and effectively second-class citizens, but not persecuted.
Some further reading on the crusades:
Jonathan Rilley-Smith. The Crusades: a History. 2nd ed. New Haven: Yale UP, 2005.
Tolan, John Victor. Saracens: Islam in the Medieval European Imagination. New York: Columbia University Press, 2002.
Brundage, James. “Holy War and the Medieval Lawyers.” In The Holy War, edited by Thomas Patrick Murphy. Columbus: Ohio State University Press, 1976.
For specific accounts of inter-religious interactions, not necessarily confined to the immediate crusade context:
Usāmah, ibn Munqidh. An Arab-Syrian Gentleman and Warrior in the Period of the Crusades: Memoirs of Usāmah Ibn-Munqidh (Kitāb Al-I’tibār) ; Translated from the Original Manuscript by Philip K. Hitti ; with a New Foreword by Richard W. Bulliet. Records of Western Civilization. New York: Columbia University Press, 2000.
The Jews and the Crusaders: The Hebrew Chronicles of the First and Second Crusades. Madison: University of Wisconsin Press, 1977.
Nirenberg, David. Communities of Violence: Persecution of Minorities in the Middle Ages. Princeton: Princeton University Press, 1996.