But if you have both gravity pulling down and the speed of the satellite pushing out, the two balance out so that the satellite can go in a circle around Earth. The first satellite to be launched into orbit around Earth was Sputnik, which was launched by the Soviet Union in 1957. Right now the farthest that we have sent a spaceship that has people onboard has been to the moon. That's only 250,000 miles away. The furthest unmanned spaceships are the Pioneer and Voyager probes that visited Jupiter, Saturn, Neptune, and Uranus.
Satellites are usually several hundred miles up.
At that distance they are too small for you to see any detail. You can see them when the sun's light reflects off the metal surfaces of the satellite. The best time to look is in the evening after the sun has set and the sky has gotten fairly dark. The ones higher up will remain there for a long time. The spacecraft that were sent out to the outer solar system, like Voyager 1 and 2, are still going.