Do massless particles follow the curved spacetime or not? I am assuming that zero (rest) mass particles don't interact gravitationally with each other and other particles. Does that mean they experience a "flat" spacetime instead of a curved one? I find it a little hard to imagine that. Also, does this mean massless particles can pass through an event horizon (not the singularity itself, I assume there would be some other way of interaction that would prohibit that).
 A: The trajectory of an object, whether massless or not, in a curved spacetime is given by the geodesic equation.
$$ {d^2 x^\mu \over d\tau^2} + \Gamma^\mu_{\alpha\beta} {dx^\alpha \over d\tau} {dx^\beta \over d\tau} = 0 $$
Solving this is a formidable problem unless there is some helpful symmetry that simplifies the working, but in the particular case of light we have a short cut because any massless particle will follow a null geodesic, and these are usually a lot easier to calculate. For example in my answer to the question Why is a black hole black? I calculate the null geodesic for light trying to escape from a black hole and show that light at or below the event horizon can't escape.
So the answer to your question is that massless particles do indeed experience the curvature of spacetime in the sense that they are deflected by it. As M. Herzkamp says in his answer, this is why gravitational lensing of light happens.
A: No, a massive body is able to bend light around it, which is called gravitational lensing. This has been observed multiple times.
EDIT
Photons are massless. Otherwise, they would not travel at the maximum speed, which is called speed of light.
Keep in mind, that gravitational lensing is not a part of Newtonian mechanics. You need general relativity for that. And in the context of general relativity, it is not mass, which exerts gravitation, but energy. And photons clearly carry energy. So, you could produce a gravitational well (even a black hole) entirely without massive particles.
A: 
I am assuming that zero (rest) mass particles don't interact
  gravitationally with each other and other particles.

That's not a valid assumption in general relativity.  Particles with zero invariant mass have energy and momentum and, thus, gravitate.  Essentially, in general relativity, the density and flux of energy and momentum are the sources for spacetime curvature.
For example, see "Do two beams of light attract each other in general theory of relativity".
