You can use Newton's laws to calculate the speed that an object must reach to go into a circular orbit around a planet. The answer depends only on the mass of the planet and the distance from the planet to the desired orbit. The more massive the planet, the faster the speed. The higher above the surface, the lower the speed. For Earth, at a height just above the atmosphere, the answer is 7.8 kilometers per second, or 17,500 mph, which is why it takes a big rocket to launch a satellite! This speed is the minimum needed to keep an object in space near Earth, and is called the circular velocity. An object with a lower velocity will fall back to the surface under Earth's gravity.
The same idea applies to any object in orbit around a larger object. The circular velocity of the Moon around the Earth is 1 kilometer per second. The Earth orbits the Sun at an average circular velocity of 30 kilometers per second (the Earth’s orbit is an ellipse, not a circle, but it’s close enough to circular that this is a good approximation). At further distances from the Sun, planets have lower orbital velocities. Pluto only has an average circular velocity of about 5 kilometers per second.
To launch a satellite, all you have to do is raise it above the Earth's atmosphere with a rocket and then accelerate it until it reaches a speed of 7.8 kilometers per second. This height is called a low-Earth orbit, with an altitude of about 200 kilometers and an orbital period of 90 minutes. With a lot more energy, you can put a satellite into a special orbit called a geosynchronous orbit. At an altitude of 42,000 kilometers, the orbital time is 24 hours, which is exactly the same as the Earth’s rotation period. A satellite in geosynchronous orbit will rotate with the Earth and appear to hang above a fixed point on the surface. Satellite television dishes all point at satellites in geosynchronous orbits — that's why they can always remain pointing in the same fixed direction.
If we increase the speed of a satellite, it goes into an elliptical orbit that extends higher above the Earth. If we keep increasing the speed, we reach a curve different from an closed ellipse, called a parabola. A parabolic orbit never curves back; the projectile keeps traveling farther and farther from the Earth. Eventually, it escapes the gravity of the Earth and goes into orbit around the Sun.
The speed required to escape completely into space from any given point is called the escape velocity. Newton's laws show that escape velocity from any given point is just the square root of two, times the circular velocity - regardless of the object's mass. So for an object in low-Earth orbit, it's about √2 × 7.8 = 11.2 kilometers per second. We can help the process by launching Eastward from the equator. There, the planet's spin of 0.5 kilometers per second acts as a slingshot to reduce the rocket’s required speed, relative to the Earth. If a rocket reached escape velocity, it would escape from Earth orbit and fly off. But even at escape velocity for the Earth, it would still be held by the larger gravity of the Sun, so it would go into orbit around the Sun. To escape from the Solar System altogether, an object would have to escape from the Sun’s gravity. From the distance of Earth’s orbit, this would require a speed equal to the square root of two times the circular velocity of the Earth in its orbit, or √2 × 30 = 43 kilometers per second. Four NASA space probes — Pioneer 10, Pioneer 11, Voyager 1, and Voyager 2 — have left the Solar System like this, never to return.