What speed do satellites orbit at?

The key point to recognise here is that an object undergoing a circular orbit around the Earth is undergoing circular motion. Circular motion is caused by a centripetal force (acting towards the centre of the circle), described by:

F_c = m_satellitev²/r   (eq 1)

In the case of an orbit, the force acting towards the centre is due to the gravity of the Earth, which is described by Newton's law of universal gravitation:

F_g = Gm_Earthm_Satellite/r²  (eq 2)

We can now equate these two expressions:

F_c = F_g => m_satellitev²/r = Gm_Earthm_Satellite/r²  (eq 3)

Now me can rearrange to make v the subject of the equation:

v = sqrt(Gm_Earth/r)   (eq 4)

This is the speed of a satellite in a circular orbit around the Earth. Note that v is proportional to 1/sqrt(r) and that the mass of the satellite has cancelled out.