Stack, good questions. Satellites range from 2,000km to 40,000km above earth. 70k km is the record. They travel more quickly closer to earth (about 90 min orbits) and go up to 24 hour orbits around 35k km which is the Geo-Stationary satellites you talked about.
On a flat earth, I imagine it looks something like this.
Satellites seem to orbit around the sun on a FE Map. See the official NASA photos of satellite orbits and the Altitude versus speed graph of satellites.
What propels a satellite to circle so fast in Low Earth Orbit is beyond me right now. Any ideas?