Is it true that the expansion of the universe means that stars in our Milky Way Galaxy are getting increasingly farther apart?
No. It means most pairs of galaxies are getting farther apart.
Within a galaxy gravity holds the stars togeher against the cosmic expansion. Stars inside a galaxy are simply too close together for cosmic expansion to "beat" gravity. Similarly, planets are held in place around their respective stars and moons around planets. Cosmic expansion overcomes gravity only on intergalactic distance scales.
Not even all pairs of galaxies are moving away from each other. Galaxies can have relative motion other than the cosmic expansion, and for instance Andromeda is moving towards the Milky Way on its way to a collision (in about 4 billion years).