I think you have two different questions here, which I'll answer one at a time. The first is "can we see light emitted right after the big bang, in the microwave spectrum?" You're thinking about this problem in the same way astronomers do. Unfortunately, we can't see light from right after the Big Bang. In this era, the universe was so hot that protons and electrons couldn't combine to form atoms. Separate protons and electrons are charged and strongly scatter light. The atoms first formed about 300,000 years after the Big Bang, in a phase known as "recombination." This is the first time that photons could travel freely through the universe without being scattered by free electrons, so these are the most distant photons we can see.
You are correct in thinking that these photons would be redshifted. Predictions of this radiation were made by George Gamow of Berkeley and Robert Dicke at Princeton in the 1950s. In 1964, Arno Penzias and Robert Wilson discovered a background noise in their radio receiver at Bell Labs. They tried to remove this noise from their system, even cleaning their antenna to remove the pigeon droppings on it that may cause this noise! Everything they did had no effect, so they called their friends at Princeton to determine if they knew what may be causing this noise. This was the first definite record of this background radiation, now known as the cosmic microwave background (CMB).
Penzias and Wilson would receive a Nobel Prize for their efforts, and today the CMB is one of the most well-studied radio sources in the universe. It is the most perfect blackbody known, with a peak temperature of 2.725 Kelvin. Significantly, three space-based telescopes have been built specifically to study this radiation. COBE (the COsmic Background Explorer) was launched in 1989 and mapped the entire CMB, finding very small fluctuations in the temperature as a function of position. This is important, because these fluctuations mean the universe wasn't exactly uniform at this time, but had slight overdensities and underdensities. These overdensities accumulated more and more matter over time, leading to the formation of individual galaxies and groups of galaxies (known as clusters). WMAP, the Wilkinson Microwave Anisotropy Probe, was launched in 2001 and built on the work of COBE, measuring the anisotropies with greater sensitivity and resolution. In 2009, Planck was launched which will be another improvement on WMAP, increasing the sensitivity and resolution of measurements. So far, all the data we have gathered fits the expansion of the universe as predicted and supports the Big Bang theory.
Your second question is "can we observe the redshifting as it happens." The answer to this is yes, but indirectly. When astronomers take a redshift measurement of a distant object, that provides a measure of how rapidly the object was moving away from us when it emitted the light. (The measure is also sensitive to how the expansion is changing.) If the expansion of the universe were constant in time, the rate at which the object was moving away from us would be directly proportional to the current distance. There would be a simple algebraic formula to calculate the distance at the time the light now reaching us was emitted. However, until the 1990s it was not known if this rate was constant in time. The conventional assumption was that the expansion of the universe should be slowing down because of the gravitational attraction between massive objects. Thus, combining very precise distance measurements of objects with redshift measurements could determine how much more quickly the universe was expanding in the past, and thus determine the mass density of the universe.
To determine distances very accurately, astronomers use "standard candles," objects that give off known amounts of light, so that the apparent brightness (which decreases as the square of the distance to the object) tells us how far away they were when they gave off the light that's now reaching us. One type of these are Type Ia supernovae. These always release approximately 1044 Joules of energy, so all supernovae of this type have approximately the same intrinsic brightness. (The estimate of that intrinsic brightness can be refined using the time-course of the light intensity.) Therefore the brightness we observe can be used to give an accurate measure of the distance to the supernova. Then, by plotting this distance as a function of redshift, we can determine the approximate rate of expansion of the universe throughout time. In the early 1990s, this was tested by a group of astronomers at Berkeley and Stanford. (See http://supernova.lbl.gov/PhysicsTodayArticle.pdf)
The important thing that they found is that the universe expansion is actually accelerating! This means not only is the universe expanding, but the rate of the expansion is also increasing with time. Today, this is usually attributed to dark energy, which is believed to make up 73 percent of the energy density of the universe. However, what dark energy actually is is not yet understood, although cosmologists are working very hard to answer this question.
For further reading, I recommend the article listed below; if you still have questions feel free to follow up!
Thanks for the question!
Ben M. (small edits by mw)
Perlmutter, S. "Supernovae, Dark Energy and the Accelerating Universe." Physics Today: April 2003. <http://supernova.lbl.gov/PhysicsTodayArticle.pdf>
(published on 12/07/11)