Can the speed of the earth, traveling through space, be determined by experiments with light?

If the speed of light is consistent, can we determine the speed and direction of the Earth, the Sun, and the Milky Way based on extremely precise experiments with light directed at different angles? (Perhaps by triangulation and the doppler shift?)

Is there a reference point in space from which relative motion is zero? (The point of the Big Bang?)

Such a point does not exist, unfortunately. One of the revelations of the Theory of Relativity is that there can be no absolute point of reference. Thus, "relativity."

We can use red and blue shift spectrum analysis to get a good idea of how fast we are moving with respect to other things in the galaxy and the Universe. All measurements must be taken with respect to something else.

We don't send out beams like radar to determine our speed. The other objects in space are too far away to get any kind of detectable reflection, (except for the mirrors we put on the moon.) We use the light that was already headed our way from all those stars and galaxies.

Since every point in the universe as we know it today was in the same location during the big bang, every point is the "center of the universe". So no there is no universal reference point.

No, it can't.

Your idea is a good one. But, far from disproving relativity, that idea is exactly what led to the discovery of relativity in the first place.

If you were in a boat, careful measurement of water waves would indeed tell you if (or even how fast) the boat is moving through the water.

A hundred years ago, scientists believed that waves must travel through a medium, and in the case of light waves the medium was called "ether". They reasoned that precise measurements of the speed of light, directed at different angles, would tell us Earth's speed through the ether. Strangely enough, all the attempts failed. Every single time anyone tried to measure the speed of light, they always got the same answer, regardless of what direction the equipment was facing. Even when the experiments were repeated several months later, when Earth had moved part way around the sun (hence would be moving in a different direction) the results always said Earth's speed through the ether was zero.

The experiments showed quite plainly that, no matter which direction you are moving or how fast, the speed of light in a vacuum will always be measured as the same number.

Einstein's conclusion from this was that there is no ether, and all motion is relative. There is no such thing as a fixed point in space. Every inertial frame of reference is just as valid as every other. This is relativity. The brilliance of Einstein's theory is how he worked out the mathematical formulas for the details.

Further experiments have verified Einstein's formulas. Relativity works.

One consequence of relativity is that you can't send information of any kind faster than c (the speed of light in a vacuum) without creating the same kinds of paradoxes which would be created by sending information backwards in time. Simply put, FTL (faster than light) spaceships are just as impossible as time machines.