I often read, that light is being bent in a gravity field and so are the paths of massive bodies (e.g. planets or stars). I also read that the curvature of space (caused by some other mass) is the reason for this. So why are planets following different paths than light. E.g. Why I don't see light orbiting the sun.
Light and matter both follow the curvature of spacetime when passing a massive object. The difference is that matter is ALWAYS slower than light, it will be in the more curved spacetime longer, so it's path is curved more than light. Compare it to throwing a ball, if you throw the ball very slowly, it will follow a sharp curve and fall down close to you. If you throw the ball as fast as you can, it will follow a much longer curve before landing. So you see that the speed changes the trajectory.
The relativistic way of looking at things is that objects don't have paths through space, they have paths through spacetime. These are called world-lines. Even in Galilean physics, the trajectory of a test particle through space is not really well-defined. For example, the planet earth doesn't have a well-defined trajectory through space that is an ellipse; it depends on what frame of reference you use. The ill-defined nature of trajectories through space gets even worse in relativity.
For test particles, these world-lines are a type of curve called a geodesic, which is defined to be straight. A ray of light always has a world-line that is "light-like," meaning that it moves at $c$. A material object always has a world-line that is "time-like," meaning that it moves at $<c$.
So both types of test particles follow geodesic paths (which are by definition straight) through spacetime, but they're geodesics that are in two completely disjoint categories.
The orbital trajectory taken is dependent on the velocity of the body. Since light always travels at $c$ and nothing with mass can, the orbits will always be different. Light does curve around the sun in a hyperbolic path. If there was something with a much more powerful gravitational field than the sun (like a black hole), light would orbit around it. This is the reason there is a ring of light just outside the event horizon of a black hole.
What general relativity describes as curved is the space-time continuum. Objects travel through this continuum on geodesics. Like a line in flat 3D space, geodesics have a direction. You don't expect two lines with a common point but different directions to continue together, do you?
Same with the world lines of your particles: The direction of the geodesic that a particle travels on depends on the speed of the particle, and as such the direction of a photon's geodesic must be different from that of any particle that moves at some lower speed.
Formally, the 4-velocity of any particle (at rest, moving, light like) is a vector that is normalized: It always has the same length. For an object at rest, the 4-velocity point only in the direction of time, for a photon it points only in a spatial direction. The faster the particle, the less the 4-velocity point into the time direction, and the more it points into a spatial directions. Consequently, the particle follows a different geodesic depending on its speed.