If the universe is expanding at the speed of light and light travels at the speed of light then how are we still able to see light from the galaxies that are moving away from us?

The universe isn't expanding at the speed of light. It's expanding at a rate of 68 kilometres per second per megaparsec. The rate of expansion isn't a speed because the greater the distance between two things, the more it's expanding.

The light from a distance galaxy actually takes no time at all. As time for a photon traveling at the speed of light time does not exist.

Experiments have been done with photons and single slits that indicate time and distance are not barriers to the photons.
A single photon going through a single slit should form a solid line.
It takes at least two photons to form a diffraction pattern. The single photon going through a single slit creates a complex diffraction pattern. This shows that that single photons go through the silt multiple times, at to the experimental observer seems to be the exact same time.

That the photons can travel multiple times through same space and time shows that photons are not limited to the speed of light, space, or time. The idea that because galaxies are moving away from us means that it could take billions of years for that light to reach us is not in agreement with the nature of photons and the Theory of Relativity.

That time is a variable and is different for different observers and doesn't exist for photons is counter intuitive to human intelligence.