It's clear the universe is expanding because light from distant bodies, like galaxies, are shifted towards the red side of the spectrum. In other words, the light measured is more red than it should be when compared to laboratory results. This is great for measuring the distance to other bodies, but can't we also measure their velocities of departure by measuring how fast the light from these bodies is becoming more and more red over time? That is, v is directly proportional to the change of frequency divided by the time elapsed?
Your understanding is correct. The doppler shift observed from a galaxy is the sum of its peculiar velocity with respect to the "Hubble flow" and the redshift due to the Hubble flow, which is caused by the expansion of the universe.
There is no direct way from a spectrum to separate these two components - they have the same qualitative result.
In principle, the expansion of the universe (or a change in the peculiar velocity) could be directly measured by looking for a change in redshift with time, which would depend on the cosmological parameters.
This is an extremely small effect and is confused by the peculiar motions of individual galaxies. Nevertheless, measuring this redshift drift is one of the prime goals of the Codex Instrument on the E-ELT (see Pasquini et al. 2010, http://esoads.eso.org/abs/2010Msngr.140...20P ) using Lyman alpha absorption systems towards distant quasars. This experiment is also planned for the Square Kilometre Array, using the 21cm line (Kloeckner et al. 2015 http://arxiv.org/abs/1501.03822 ).
In both cases, to overcome the experimental uncertainties (eg at 21 cm, it amounts to line drifts of 0.1 Hz over a decade), then observations of millions of galaxies must be combined.
There is no prospect of measuring this effect in an individual galaxy, furthermore I fear your understanding of cosmological redshift is flawed. The dependence on distance is a statistical average, not an absolute dependence. Individual galaxies are moving in individual gravitational potentials from objects around them. This gives them their peculiar velocities with respect to the flow. This velocity could increase or decrease as a galaxy got further away, but is never expected to be large enough to be detectable on human timescales for any individual galaxy. In addition, any change in peculiar velocity should average to zero when looking at millions of galaxies, leaving the redshift drift due to the expansion.
When light is emmitted by an object that is moving away from us, the light is shifted to the red. This is an example of the doppler effect. So the red shift measures radial velocity, not distance.
By estimating the distance to galaxies, Edwin Hubble found that the velocity of galaxies correlates with their distance. So by measuring red shift you can estimate the distance of a galaxy.
The distance is only an estimate, and there is cosiderable uncertainty in our knowledge of the distance of galaxies. Hundreds of millions of light years of uncertainty. The distance to even the fastest moving galaxies doesn't change fast enough to be measured. Even if a galaxy is moving close to the speed of light, in one year it would only have travelled 1 light year, much less than the uncertainty in the estimate of the distance.