You don't feel acceleration. When onboard the ISS, you are accelerating towards the earth (down) due to gravity: if you didn't, you would just fly away from the planet. Because you and the ISS are accelerating exactly the same way, you don't feel a thing. You don't feel a force if it's accelerating you: you feel pressure caused by opposing forces. Here on the ground, I feel the floor beneath my feet opposing my normal gravitational acceleration.
If you run a thruster on the ISS, then the ISS starts accelerating differently than you do, and eventually, one of the walls is going to collide with you. Then you'll feel that wall interfering with your own gravitation acceleration, and feel something like weight.
Light undergoes acceleration due to gravity: look up 'gravitational lensing' for that. To understand how light can accelerate with a constant speed, you have to understand the difference between speed and velocity, and what acceleration really means.
Speed is a 'scalar', just a number with no direction. If you're travelling 30 KPH, that's your speed.
Velocity is a 'vector', a number with a direction. Driving 30 KPH north is much different than driving 30 KPH south: clearly, you'll end up in different locations regardless of your speed.
Acceleration is not a change in speed, it is a change in velocity. Think of a car. There are usually three ways to accelerate a car. To increase your speed (scalar), step on the accelerator, and you'll feel your seat back pushing into you harder as it accelerates you with the car. To decrease the speed (scalar), hit the brake and you'll feel your safety straps accelerating you with the car.
But what happens when you turn? Your speed stays roughly the same (exactly the same if you're skilled enough), but you are changing your direction. Your 30 KPH north is becoming 30 KPH west, and the change in direction is an acceleration. Depending on whether your car is build to drive on the right or the left, you'll have a tendency to either push against your door or into your passenger's lap. That's still acceleration.
If a photon is passing by something heavy, it will be accelerated towards that object, changing its course but not its speed. If a photon is going towards or away from something heavy, it can't properly accelerate by changing speed. I'm not a physicist, but I believe that it increases or decreases energy by changing its frequency. In other words, things that you would expect to increase its speed will instead increase its frequency ('blue-shifting' it if it's visible light), and what you would expect to decrease its speed will instead decrease its frequency ('red-shifting' if it's visible light).