Most Muslims do not blow stuff up. I took one class of Shiite Islam in college, which covered the origins of Islam. I also studied Arabic. I spent many, many years in a predominantly Muslim country. My wife was not raised Muslim, but I have some Muslim in-laws. My children might be descended, physically, from Muhammad. I'm a little unclear about whether that historical imam of Indonesia that my wife, as an adult, was told that she was descended from, was really descended from Muhammad.
So I have more experience with Muslims than the average American.
What I was told, at least, in college and elsewhere, is that the Al Qur'an has three categories of people, kafir, people of the book, and Muslim. Muslim means submitter. They use that for someone who accepts Islam (submission, the name of their religion.) To become a Muslim, one recites their saying about belief in one God and their belief that Muhammad is his prophet. There are also five pillars of Islam, including going to Mecca.
There is a concept of 'jihad'-- the lesser jihad is war, the inner jihad is the struggle with self.
Islam is not supposed to tolerate idolatry, or 'shirk' associating partners with God. So pagans who reject Islam are kafirs- (I forget the plural in Arabic) that is, disbelievers, and Islam is really against them. The other category is for pre-Islamic religions that have a holy book, so Jews and Christians are considered to be 'people of the book.' Islam is generally more tolerant of them, allowing them to exist if they pay a fine/tax.
Incidentally, I hear the Ottoman empire charged Christian was less than Byzantine taxes of the same people, which may have made it easier for them to maintain control.
There are some passages of the Al Qur'an where Muhammad was at war with Jews. There was a Jewish kingdom down in the Arabian peninsula during his day. Apparently, a number of Arabs had converted to Judaism. So some of the anti-Jew rhetoric comes from the rhetoric in Islam during a time of war. Moderate Muslims take this as context specific for when they were at war. Some of the radical Muslims take the anti-Jew stance as a permanent thing.
The Al Qur'an also says Christians worship God, the Son, and Mary. Muslims are against bowing down to statues and do not even draw pictures of Muhammad's face. So they do not like some of the veneration and iconography practices.
The Al Qur'an also says good things about the Torah, Gospel, and Psalms, and says Christians should read these books to determine if the Al Qur'an is true. It actually makes room for a lot of 'ammunition' for discussion with Muslims by this and other passages.
The Al Qur'an calls Jesus, or 'Isa, the Word of God and the spirit of God. This is a starting point to tell Muslims about Jesus as the divine Logos.
I don't want Muslims blaspheming the Holy Spirit, so if I have encountered one who seems like he might want to go the route of calling Muhammad the Paraklete, I might say I consider the Holy Spirit to be divine and considering calling the Paraklete a man to be 'shirk.' Challenging one who does so with 'shirk' might shut that part of the conversation down.
Some Muslim, Sunni, I think, tend to think of Gabriel as the holy spirit. They are confused about the concept of the holy Spirit.
Most Muslims are Sunni. A minority are Shi'ite/ Shi'i. After Muhammad died, they kept choosing leaders, originally based on their closeness to Muhammad. Muhammad's surviving descendants all came through his daughter Fatima and cousin/son-in-law 'Ali. The community made one of Muhammad's fathers-in-law, his best friend 'Abu Bakr the first Calif-- ruler of the Islamic community-- after he died.
So a Calif died and passed it to his son, and the descendants of Muhammad and his followers would rally and fight. There was a civil war where one of Muhammad's wives, Aisha, the one he married when she was maybe 6 but did not have sex with until she was maybe 9 or so, was on one side and Muhammad's grandson was on the other. After a series of skirmishes, both Hussein and Hassan, Muhammad's grandson's were wiped out. Their lineage became, for many generations, the leaders of the Shi'ite community.
Muslim beliefs are based on the Al Qur'an, which they consider to be spoken by God, and also to a lesser extent, hadith, sayings of Muhammad reported by different people. Shi'ite Islam has extra hadith.
