1. Mecca (Saudi Arabia) is by far the most important, and the site of the annual Hajj (pilgrimage), which all Muslims are expcted to undertake once if they are physically/financially capable of making the trip. Muhammad (pbuh) came from there, as did all of the first Muslims who heard the message of Islam from him. The Kaaba, which Muslims face during prayer, is located in Mecca. We believe that it was a shrine built by Abraham, which later fell into disuse and was filled with pagan idols. When the Muslims took control of Mecca they removed the idols and restored its original status as a holy site for Abrahamic monotheism.
2. Medina (Saudi Arabia) is the second most important, and people often take a detour to visit it after their pilgrimage to Mecca. The first Muslims were persecuted in Mecca and had to flee. Medina (originally a Jewish settlement called Yathreb) was the only place where they could find refuge. About a decade later, they returned to Mecca and it is from this point on that the Muslim community became a political entity as well as a religious group.
3. Jerusalem (Israel/Palestine) is the third major holy city. As I said before, the first Muslims lived alongside Jews. One of the first things that God told Muhammad (pbuh) is that Jews, Christians, and Muslims share the same God. So in the early years, when Mecca was a pagan stronghold instead of a Muslim one, Muslims prayed facing Jerusalem, alongside the Jews. They did not face Mecca until the Kaaba was purged of its pagan idols. Muslims also believe that Muhammad (pbuh) ascended to heaven for one night and was given a glimpse of it during his lifetime. Jerusalem was the point from which he ascended. This is why Jerusalem is so heavily disputed in the Palestine/Israel conflict. It is an important site for both Jews and Muslims, so both groups want control of it, or at the very least unrestricted access to their major shrines. Pro-Israel groups will claim that the Muslim claim is invalid because the Qur'an does not specifically mention Jerusalem. But I have explained that Jerusalem is important to us because it is the site of a major event in Muhammad's (pbuh) life. The Qur'an is our guide on how to live. It is not a history text or a biography of Muhammad (pbuh). So no matter what an extremist on any side tries to say, both Muslims and Jews have valid reasons to consider Jerusalem a holy site and demand the right to be there, although it is the #1 site for Jews and only the #3 site for Muslims.