What is the speed of the planet earth at its slowest and fastest in its orbit around the sun?

Use the formula for speed of an elliptical orbit:

v=\sqrt{GM\left({2\over{r}}-{1\over{a}}\right)}

for apogee and perigee radii r, semi-major axis a, and solar mass M.

Should give you the answer you're looking for.

What is the value for G, what is the value for M, what is the valuefor r, and what is the value for a? I was looking for answers like 20,000 miles per hour and 22,000 miles per hour.

You can Google that.

G is the universal Gravitational constant, which has a value of 6.67\times 10^{-11}
The Mass of the sun is roughly 2.0\times 10^{30}
You have to Google for the apogee and perigee 's values and for the value of the length of the semi-major axis.

From:
Curious About Astronomy: How fast does the Earth go at perihelion and aphelion? (http://curious.astro.cornell.edu/question.php?number=614)

They use the same equation as Capuchin provided to show that the Earth's orbital velocity around the sun is about 29,300 m/s at aphelion (when the earth is furthest from the sun) and 30,300 m/s at perihelion (when the earth is closest to the sun).