space time - When you travel fast around a black hole, do you experience high speed yourself?

If you jump out an aeroplane, do you experience high speed? Many people would say you do, since your velocity relative to the ground increases.

Note the importance of the phrase "relative to the ground" since to describe speed you always need to say what you are measuring the velocity relative to.

A simple choice is to measure your velocity relative to a distant observer. In this case then, yes as you pass close to the black hole your velocity relative to the distant observer increases. As you approach the event horizon, you velocity relative to the observer will approach the speed of light.

Depending on how close you pass the the event horizon, you will experience significant time dialation, so that more time will pass in the observer's frame than in yours. This means that if you manage to escape the gravitational pull of the black hole, and return to the observer, you will have found that the observer has aged compared to you.