There are numerous conflicts in "Kim", but they can be generally summarized according to their domain; physical or spiritual.
The physical conflict, i.e. that taking place between people via words and actions, is the "Great Game", the geopolitical machinations of the Russian and British Empires for control of Central Asia. "Kim" is credited with popularizing this conflict, as it was playing out at the time of Kipling's writing. Much of the action is depicted through Kim's perspective, which is hardly omniscient, but we do get a sense that there are much larger things going on. Kim is basically acting as a spy and courier for the British throughout the novel, although this job is complicated by the variety of people, cultures and politics intersecting in Central Asia.
The spiritual conflict, i.e. the one taking place internally, is the constant juxtaposition of the Western and Eastern worlds, and the question of where Kim's identity falls between the two. Kipling represents the choice as a difficult if not impossible one, as the Western forces are not portrayed as villains, nor are the Eastern forces represented as simpletons or backwards mystics, as stereotypes might predict. The novel ends without a resolution to this conflict; Kim is clearly torn between his personal allegiance to the lama, and his unshakeable physical appearance as a sahib.
Thus there are two main conflicts: the cold war for central asia, and the question of personal identity.