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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 in 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.

To help with James K's excellent answer, a visual representation might help. Let's look at a thought experiment - Newton's Cannonball.

Let's say you have a cannon, high enough that it's being held above Earth's atmosphere.

You fire it, and it falls to Earth a little ways away ("D" in the below diagram).

You fire another one with more power so it's moving quicker, so that it falls to Earth farther away. ("E" in the below diagram)

Eventually, you fire a cannonball with such extreme velocity, that it's "falling" around the Earth fast enough that it never reaches the ground.

This is orbit. Orbit isn't necessarily being really high up and moving slowly. More often than not, orbit is going sideways fast enough that you're falling without losing height.

So, why on places like the ISS, does it look like things are just floating around? Let's quickly go back to the cannonball.

Imagine the cannon fires two cannonballs at the same time, both going fast enough to be in orbit. These cannonballs are going blisteringly fast... but they were fired at the same time, going the same speed, so they stay together. If you can imagine being one of the cannonballs, the other cannonball would look like it's just floating next to you as you fly around the Earth. This is because, relative to each other, the cannonballs have next to no relative velocity.

The ISS, similarly, is traveling about 7.66 km/s, or around 27,600 km/h (about 17,150 miles per hour for those using imperial measurements). It's going fast. But when you're on there, everything is going the same speed, because you're all on the ISS together.

So if you let go of a pen on the ISS, it's still traveling the same speed as you - around 27,600 km/h. But because it's going the same speed as you, relative to you it just looks like it's floating.

Earth isn't just floating in orbit around the Sun, it's in orbit at (on average) 107,000 km/h (or 67,000 miles per hour).

Our Solar System isn't just floating around the center of our galaxy, it's in orbit at around 828,000 km/h (or around 514,500 miles per hour).

These are all hard to comprehend speeds - we can all agree though, that this isn't just floating around. Things are moving fast.

Things can appear to be just "floating" because their relative velocity to the observer is small. But I hope this all gives an explanation of how just because something appears to be slowly floating around, that doesn't mean it's not still moving fast from a different point of view.