The context
The Qur'an speaks to other traditions from the inside out.
One of the most important things to understand about the Qur'an's engagement with other faiths is that it does not treat Judaism, Christianity, and other monotheistic traditions as alien or hostile. Within its own account, it treats them as earlier expressions of the same prophetic line — originating from the same source, though with points of departure the Qur'an identifies explicitly. This is a starting point worth understanding before engaging in any serious comparative study.
The Qur'an addresses the Ahl al-Kitab — the People of the Book — directly and at length. It affirms shared prophets, shared stories, and a shared moral framework. It also identifies specific points of departure. Understanding both is essential for any serious comparative engagement with what the Qur'an actually says.
JudaismShared prophets, distinct theology
ChristianityJesus in the Qur'an; Trinity and its critique
Earlier revelationTorah, Psalms, Gospel — Qur'anic view
Shared moral frameworkJustice, accountability, care
Points of departureWhere the Qur'an disagrees, and why