Randee, the easiest place would be right over the equator, going west at about 6pm. Altitude and airspeed would be related (the higher the altitude, the higher the airspeed). You'd need to be going about 1700 km/h, since you need to go at least 40 000 km (circumference of the earth) in 24 hours.

Cheers

Stephen gave the only constant-speed "great circle" solution, following the equator at about Mach 2.2. However, there are many other solutions, including some possible for non-supersonic planes. For instance, near the spring or fall equinox even a small plane could fly around the North Pole in a circle and do the same thing. Theoretically if you were right by the Pole you could walk (except your head would be too high, and the Sun would be skimming along the horizon so that "setting" would not be apparent) Any point on Earth can be the starting point for some solutions. Any altitude will work (with minor adjustments to airspeed)

Good Hunting!
RD