How fast is Earth moving through the universe? As the galaxy is moving and the solar system orbiting the galaxy and the Earth orbiting the sun. So how fast is each object moving and what is the fastest we move at?
Do we even know how fast the galaxy is moving that is not relative to another galaxy (although I guess velocity has to be measured relative to something).
 A: Solar system is moving at $368\pm 2$ km/sec relative to the microwave background in the direction towards the constellation Leo.
A: Velocity does indeed have to be measured relative to something. We can measure our radial velocity relative to any other astronomical object we care to, by measuring Doppler shifts. But if you want to know our velocity "relative to the Universe as a whole" rather than relative to any one object, we have to be a bit careful to define our terms.
Because the Universe appears to be approximately homogeneous and isotropic, it makes sense to define a "rest frame" at any given point. (The rest frames at different points are moving with respect to each other -- that's what it means to say that the Universe is expanding.) This "rest frame" is essentially the frame in which the stuff surrounding that point appears to be moving isotropically (the same in all directions). In practice, the best way to define that rest frame is to find the frame in which the cosmic microwave background appears the same in all directions (has no dipole moment, to be precise). Relative to this frame, the local group of galaxies is moving at about 600 km/s (Wikipedia gives precise numbers and probably citations that I'm too lazy to look up).
People sometimes worry about whether the existence of a preferred "rest frame" of this sort is in conflict with the principle of relativity. The answer is that it isn't. There are a couple of ways to see why. One is to note that the principle of relativity says that the laws of physics have no preferred frames, but particular solutions to the laws can have preferred frames. Another way of putting it, which I prefer, is that the "rest frame" we use in cosmology is simply the center-of-momentum frame of a bunch of particles (namely the CMB photons in our neighborhood). In other contexts, we're not surprised or worried by the fact that a bunch of particles have a rest frame, so why should we worry about it here?
A: The Earth is moving by 30 km/s around the Sun and relatively to the Sun. The Sun is orbiting the center of our Galaxy, the Milky Way, by the speed of about 200-250 km/s. Our Galaxy is moving relatively to the Local Group where it orbits and the Local Group falls toward the Virgo Cluster of Galaxies.
However, the latter two velocities are small relatively to the Local Group's velocity relatively to the cosmic microwave background, the closest approximation of "the right frame" that cosmology may offer you: the Local Group is moving by 600 km/s relatively to the cosmic microwave background. That's 0.2% of the speed of light.
At any rate, it's important to appreciate that all velocities are relative, all observers who are moving by constant velocities relatively to each other may use the same laws of physics (the principle of relativity), and the cosmic microwave background is so weak that it doesn't break the democracy between various reference frames in a significant way.
A: First you have to think about what you mean by space. The concept of how fast you are moving through the universe sounds like you are assuming an absolute space as envisioned by Newton rather than a relational space as promoted by Leibniz. So, as worded, it seems that you need an absolute background spacetime, which is not the case with special relativity. The space described by inertial reference frames in areas of small curvature is usually used as "background space" and, as Ted says, descibes an isotropic and homogenous space. Motion is only relative to other inertial frames, and if you choose a rest frame, you have to define one yourself.
A: Earth spins, Earth orbits the Sun, the Sun orbits the galaxy, the galaxy is moving and the universe is expanding. How fast is Earth moving through the universe? Speed has to be measured in relation to something, so where is the point of relation in the universe? If there is something like the center of the universe far beyond our observable universe, this could be used as a point of measure. We don't know if the center of the universe exist, but if it does, the whole universe could spin around its center and the centripetal acceleration could cause it to expand, and we could have accelerated expansion of the universe without this mysterious dark energy which is the most popular explanation today. If there is something like a Higgs field which make mass out of energy, this could also be related to the center of the universe. And if something like a universal God exists, he probably have a nest in the middle.
But even if there is such a thing as the center of the universe, it would be equally valid to say that the Earth is standing perfectly still while the center of the universe is moving. Then we are all standing still in the center of our own observable universe.
A: I don't buy into the idea that "one's speed at the equator is cancelled out because half the time you are spinning in the opposite direction" (in relation to the sun). You are still going 1040 mph, though it can be argued that this speed is cancelled out during half the day if the Earth is moing around the sun in the opposite direction you are travelling on the surface of Earth.. The Earth travels around the sun at 66, 666 mph. The Sun (our solar system) rotates around the center of the Milky Way at beween 420, 000 and 540, 000 mph. Finally, it is believed that the Milky Way is traveling or moving around a "local group" of galaxies at 2, 237, 000 mph. So, do the math. We are .making pretty good time even when we feel as if we're standing still. AND if we are in high-performance sports cars, that is just a little more math.
