I know that Earth orbits the sun because it has initial momentum and that prevents it from falling into the sun. But does moon that orbits the Earth also have initial momentum?
That's how we mostly think of it. However, the situation can arise where you get a so called 'horse shoe' orbit where the orbit of a small satellite follows a horse shoe shaped path which lies near to the Earth's orbit. There are a number of asteroids that actually do this. See this Wiki link. It is a moot point whether you would consider the satellite as orbiting around the Earth or the Sun. This could be another example of where insisting on classifying the orbit as one or the other is more trouble than it's worth.I aways imagine the earth and moon as a single item orbiting the sun