Why does physical entities travel in straight paths in a flat space-time and in geodesic in curved spacetime? Is it due to Inertia? If it is, then why does waves also follow the same pattern?
A free-falling particle follows a straight line independently whether a spacetime is flat or curved. What matters is how you define a straight line!
A straight line is defined as a curve which parallel transports its tangent vector.
In a flat spacetime (Minkowski) that definition matches with the intuitive concept of a straight line as the path of a light ray in our ordinary three dimensional space. In a curved manifold it is a geodesic obeying to the parallel transport constraint.
The geodesic equation is
$(D/d\lambda) (dx^\mu/d\lambda) = (dx^\nu/d\lambda) \nabla_\nu (dx^\mu/d\lambda) = 0$
$D/d\lambda$ directional covariant derivative
$dx^\mu/d\lambda$ tangent vector
$\nabla_\mu$ covariant derivative
$\lambda$ affine parameter
Note: Both a massive particle and a massless one (photon) follow a geodesic.