Finding the cosmological redshift of a galaxy in the expanding Universe Firstly, I understand what the Doppler effect is when it comes to sound or light waves.
From everything that I've read, we are told that the universe is constantly expanding since the all the radiation we observed is red-shifted.
Assuming we are observing a distant galaxy/start that is moving away from us, the EMR waves incident on us from that galaxy is red shifted.
My question is:
How do we know that the light is red shifted? When you measure its wavelength  you are given just ONE value $\lambda$, right? How does one know what its original wavelength was to begin with? Its only after we know both the values (the actual wavelength when it was emitted and the wavelength that we measure hear on earth) that we can claim that the galaxy/star is moving away.
I always assumed that one would observe both the wavelength of the photons and the energy and there is some sort of disparity there that tells us the light is red-shifted. But that does not make sense since one usually determines the wavelength based on the energy of the photon......I am confused here.
 A: Every star or galaxy contains some elements, and each element emits a particular frequency. Here are the lines of the Sun (https://en.wikipedia.org/wiki/Fraunhofer_lines)
In particular, Hydrogen is present almost everywhere and Hydrogen lines are visible in most galaxy spectra. The Hydrogen-alpha line is particularly strong in many galaxies.

This electromagnetic radiation is at the precise frequency of
  1420.40575177 MHz, which is equivalent to the vacuum wavelength of 21.1 0611405413 cm in free space.

(https://en.wikipedia.org/wiki/Hydrogen_line) (http://www.astro.washington.edu/courses/labs/clearinghouse/labs/HubbleLaw/measurements.html)
They simply compare the standard value of the H-line ( or of any other element) with the one coming from the star/galaxy, and get the value of z (the redshift): $1 + z = \frac {\lambda_{obsvd}}{\lambda_{emit}}$ , $$z = \frac{\lambda_{obsvd}} {\lambda_{emit}} -1$$.
A value of 211 cm would give a redshift (211/21.1 -1): z = 9
Update

But that does not make sense since one usually determines the
  wavelength based on the energy of the photon

Such high frequencies cannot be detected. Usually it is the other way round, but you are right: there is only one wavelength that corresponds to a frequency, and that never changes

I have a followup question. Does one measure the energy of the photon
  incident and then calculate what its "observed" wavelength is?

Spectrography measures directly, as I said, the wavelength of the radiation,  (https://en.wikipedia.org/wiki/Spectrography)
if it is 211 cm., you know right away the cosmological redshift (z) = 9
A: The answer to this question is that if you can only see one line or feature in the spectrum then the redshift cannot be measured unless you have some other information that leads you to guess what the line or feature in the spectrum is due to (e.g. the 21cm line of hydrogen at radio wavelengths is so strong and ubiquitous it can usually be identified immediately).
The more usual situation, especially in the optical and infrared parts of the spectrum, is that you have two, or often several more, features or lines in the observed spectrum.
If we have two lines with wavelengths in the rest (laboratory) frame $\lambda_1$ and $\lambda_2$, and say these are redshifted by an amount $(1+z)$, where $z \simeq v/c$ (approximately true when $v \ll c$). We label the observed redshifted wavelengths from the distant galaxy as $\lambda_1^{\prime}$ and $\lambda_2^{\prime}$, such that
$$ \lambda_1^{\prime} = (1+z)\lambda_1\ \ \ \ \ \ \ \lambda_2^{\prime} = (1+z)\lambda_2$$
The point of the algebra is that all line wavelengths are shifted by exactly the same factor $(1+z)$. Hence a pattern of lines in the spectrum (e.g. the Balmer series of Hydrogen, or a close pair of calcium H & K lines, or sodium D lines) is replicated and can be recognised easily as such in the redshifted spectrum. Then, with the lines identified, the redshift can readily be calculated from the observed wavelengths.
