I think the best way to understand both of these concepts, is to first move away from Newton's view of gravity and understand it from Einstein's perspective.

Einstein's theory of gravity is general relativity.  It explains that gravity is actually curved spacetime; it's not exactly a "force" like Newton postulated.  When a mass (a planet, a star) occupies space, it warps that space (and time) around it, causing objects to follow paths (orbits) in this curvature.

Think of a bowling ball sitting on a trampoline.  The bowling ball is the mass and the trampoline is space.  When you place the ball on it, the trampoline net depresses.  If you roll a marble on that trampoline, the marble's path would be deflected as it neared the bowling ball.  Would you say the bowling ball is really pulling the marble toward it?  Or would you say the marble is moving toward the bowling ball because it is following the only path it can, given the curved netting that the bowling ball creates on the trampoline?  According to general relativity, it's the latter.

A black hole creates such a "depression" in spacetime that something would need to travel faster than light to escape the severely warped space around it.  Nothing can do this -- not even light -- hence the name "black" hole.  Now, the concepts of black holes and curved spacetime did not exist in Newton's day.  I do not know if Newton believed light had mass, but you see where I'm going with this...

Curved spacetime negates gravity being a "force," so something does not need to have mass to feel gravity's effects.  We often assume something has to have mass to experience a force because of Newton's laws (e.g., F=ma).  This is not exactly the case.  Newton's laws, as Einstein showed, are limited in their scope of applicability.  They work great for everyday mechanical and astronomical uses (predicting where a trajectory lands, the motions of the planets, sending astronauts to the moon).  But Newton's laws break down when dealing with very large masses (black holes) and relativistic effects (speeds at or approaching the speed of light).

Light is massless.  It has energy, but no mass.  Still, light exists in space.  So it is subject to the curved spacetime paths that masses create.  The light around a black hole gets "pulled" toward it not because the black hole is reaching out and exerting a "force" on the light, but because the light has no choice but to follow the curved space around the black hole.  Just like the marble you rolled on the trampoline has no choice but to follow the curved path around the bowling ball.

Light also does not accelerate to higher speeds as it nears a black hole.  Light always travels at c=299792458 m/s, never more, never less.  The speed of light c is a universal constant.  If light did accelerate as it neared a black hole, it would be able to escape the black hole's curved spacetime and the black hole would no longer be "black."

Consider this link to help answer your question.

I hope this helped make sense of everything!