How fast is the planet earth rotating on its axis?
How fast is earth traveling around the sun?

When its speed is measured at the equator, the earth is rotating at
1,532 feet per second. The planet moves around the sun at more than 18
miles per second.

This information is expressed in various manners (and differing levels
of precision) on a number of websites. Here's a selection of sources
for you:

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"The Earth rotates on its axis at about 1,100 miles an hour, a motion
that creates day and night. The Earth orbits the sun at about 67,000
miles an hour, a motion that takes one year."

Houston Chronicle
http://www.chron.com/content/interactive/space/astronomy/news/1999/ds/990602.html

==============================

"Earth rotates on its axis at almost 1100 miles/hour. Earth moves
around the sun at over 68,000 miles/hour."

Howe High School
http://www.howe.k12.ok.us/~jimaskew/pmotion.htm 

==============================

"In other motion sickness news, the Earth rotates on its axis at about
1,100 miles per hours, orbits the sun at about 67,000 miles per hour,
and circles the Milky Way at about 486,000 miles per hour. "

WineX
http://www.winexwired.com/archives/steve69.htm

==============================

"The Earth's surface rotates about its axis at 1,532 feet per second
-- slightly over 1,000 miles per hour -- at the equator, and the
planet zips around the sun at more than 18 miles per second."

Space.com
http://www.space.com/scienceastronomy/solarsystem/earth-ez.html

==============================

"Earth goes around the sun at a brisk 107,000 kilometers per hour
(66,000 miles per hour)! If our interplanetary spacecraft is aimed in
the same direction Earth is already going, it will get a big head
start. Also, Earth rotates eastward on its axis, one complete turn
each day. At the equator, Earth's surface is rotating at 1675
kilometers per hour (1041 miles per hour)!"

The Space Place (NASA)
http://spaceplace.jpl.nasa.gov/ds1_mgr.htm

==============================

"The Earth's surface is rotating about its axis at 1,532 feet per
second at the equator, and the planet zips around the sun at more than
18 miles per second!"

Maryland Science Center
http://www.mdsci.org/shows/imax/spacestation/factsheet.cfm

==============================

My Google search strategy:

"rotates on its axis" + "miles per hour"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=%22rotates+on+its+axis%22+%22miles+per+hour

"rotates on its axis" + "kilometers per hour"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=%22rotates+on+its+axis%22+%22kilometers+per+hour

I hope this is helpful. If any of the links do not function, or if
anything I've said is unclear, please request clarification before
rating my answer, and I'll gladly offer further assistance.

Best wishes,
pinkfreud

Thank you.  You have been very helpful!

Thank you very much for the five-star rating and the tip!

~pinkfreud

Note that the speed at which the Earth orbits the Sun is not constant.
 Kepler's laws tell us that all such orbits are elliptical, with the
centre of gravity of the system at one focus.  When the Earth is
closest to the Sun, it is travelling fastest; when it is most distant,
slowest.

Note also, that the linear, surface speed of the rotating Earth
depends on latitude.  At the poles, the Earth is not rotating at all
in terms of m/s or whatever.  Maybe the best, but most trivial, answer
to this part of your question is "once per day".