Curved paths through spacetime when standing still? I have heard that falling objects fall at the same rate irrespective of their mass. They are 'following straight line paths through curved spacetime'. Does this mean that objects that accelerate in any other way, such as a rocket taking off, are following curved line paths through spacetime?
Specifically, what path is being followed when stationary on the surface of a planet?
 A: 
Does this mean that objects that accelerate in any other way, such as a rocket taking off, are following curved line paths through spacetime?

Yes, but we should be clear on what we mean by "curved." The natural notion of a "straight line" in spacetime is that of a geodesic. This is a path that parallel-transports its own tangent vector. Physically, it is the path that always goes in the direction most aligned with "forward." If you walk on the surface of the Earth in a "straight line," you will end up following a geodesic on the surface (known as a "great circle" to cartographers).
Anything that is not a geodesic is "curved" for our purposes. This may seem weird, because standing "still" on the Earth's surface has a very easy-to-write-down worldline in spherical/Schwarzschild coordinates: $r$ is the constant radius of the Earth, $\theta$ is the constant co-latitude, $\phi$ is the constant longitude, and only $t$ changes. But just as a straight line in $\mathbb{R}^2$ has a complicated formula in polar coordinates, a geodesic can have a complicated representation in a given coordinate system, and conversely a simple-to-describe worldline can fail to be a geodesic.
