Why do things float in space, though the gravity of our star is always present? In the solar system, things float around. We are so certain that the gravity of our sun exists. Still, why does that gravity not influence satellites and other objects there?

Our solar system is also orbiting the center of our galaxy, and the gravity of our black hole, but still many objects float around. I wonder why.
It is not true that "objects float around" in the solar system.

Perhaps you have seen video from the space station, and you can see things floating. This is not because there is no gravity, but because everything in the space station going at the same speed in the same direction. This makes it look as if things are floating. In fact the space station and everything in it, is moving at about 7km per second around the Earth.

The Earth and everything on it is is also moving at 30 km/s around the sun. The satellites around the Earth are also affected by the sun's gravity, but because they are affect the same as the Earth, they are also moving at about 30km/s around the sun. If two things are moving at the same speed and in the same direction, it will look as if they are floating. In fact they are "whizzing".

The Sun, Earth and satellites are also whizzing around in the gravity of the galaxy.  The black hole is only a very small part of this; most of the gravity of the galaxy is in the dark matter that we can't see.  But we don't feel this pull because the Sun, Earth, the satellites and you are all being pulled at the same time.