I'm curious to know, does the sun really revolve around the black hole at the center of the galaxy?
Sun revolves around the center of our galaxy. It takes about 225-250 million years (the timespan known as galactic year) to complete full circle. Taking this image from the wiki
The center of the galaxy indeed contains the supermassive black hole. However the situation is significantly different from say the solar system. In the solar system almost all the matter is contained in Sun. The rest of the planets, moons, asteroids and comets contributes very little. So planetary orbits are determined mostly by Sun's gravity. In contrast the supermassive black hole in the center of our galaxy constitutes a miniscule proportion of the total galaxy mass. Also the significant part of the galaxy mass is in the dark matter halo which is spread even more than a visible matter. So saying that Sun revolves around supermassive black hole in the center is probably misguiding - its motion is determined mostly by other matter gravity.
Geometrically the supermassive black hole , Sag A*, is at the centre of the galaxy. However, its 4 million solar masses contribute only a small part of the gravitational force of the billions of solar masses that keep the Sun in its orbit. Therefore, dynamically speaking, the sun does not revolve around Sag A* but around the total of this black hole of 4 million solar masses and billions of stars.