There are many ways to do spectroscopy because of the wide range of wavelenghts of light. I won't go into detail, but essentially what spectroscopy does is either:

1. Put energy into a sample and see what is absorbed (absorbtion spectroscopy)
2. Put energy in a sample and see what comes out (emission spectroscopy)

The reason those two methods produce characteristic results for each element is the following: An atom is made up of a nucleus of a certain charge and electrons canceling that charge around it. Those electrons are confined to so-called orbitals due to quantum weirdness (the "quantisation" of the orbitals is literally the origin of the word quantum). Those orbitals have different energies (you can imagine that an electron being very close to the nucleus is more strongly attracted than an electron which is farther away).

Because the electrons need to always be on those orbitals with fixed energies, only certain energies of photons can interact with them (if a different energy photon wanted to interact with an electron it would need to push the electron "between" two orbitals which is forbidden by quantum mechanics)

So now only certain energies of photons (which relate directly to wavelength) are absorbed, the rest passes uninterrupted leading to bands in the spectrum where lots of photons are absorbed.

Now depending on how many electrons your atom has and how far away they are from the nucleus those absorbtion bands will vary, giving you a good idea which atom you are looking at.

Emission spectroscopy works the other way around, instead of you seeing what is absorbed, you randomly put energy (often using heat) into the atom. When the atom wants to go back to its most stable state it has to emit a photon, this photon needs to correspond to a gab between two orbitals (because else the electron either starts or ends outside of an orbital (which is forbidden))

Waiting for an actual spectrographer to weigh in, but I think there are databases of molecular emission spectra that can be used to match a sample for complex molecules.

Each element has a known set of emission lines. Mixing elements together in molecules can shift these lines some and add them together. https://en.wikipedia.org/wiki/Emission_spectrum has some examples across the visible spectrum.

Afaik emission spectra are measured for astronomy and passive remote sensing (since generally you're just capturing what's already being emitted). Most spectrometers or spectroradiometers can be used to measure emission or reflection, so then it's just a question of if you want to measure a sample's reflectivity by shining a known source of light at it or it's emission by exciting it with heat or electricity or lasers.

I think Raman spectroscopy is used to excite a crystal lattice with a laser to identify its structure based on the wavelengths emitted outside the laser band, so it has a specific application on crystallography, just like X-ray diffraction.

Also don't forget mass spectrometers literally rip apart molecules and sort the atoms by mass, so the relative composition of elements can be measured directly.

What drives the choice of method?

Not all methods can pick up on all materials, and some methods are easier to use or faster or more accurate or you can so multiple things at once or probes something else than just what the material is.

For instance, EDX is a method where you hit the sample with an electron beam from a scanning electron microscope (SEM) and measure emitting x-rays. You can probe lots of elements with this, but not the very lightest of elements (skipping couple at the beginning of the periodic chart, don't remember which). It is very accurate in terms of spectra and because it is an electron beam it has very high physical resolution. However, you need to use pretty high energies which can destroy or modify the sample when doing the scans. Also, you need an electron microscope with the capability, it is common to skip it and use the money for improving the electron microscope in other means. And then you need to put your sample in a high vacuum, which might be a problem. It is not exactly fast and SEMs are used a lot so the tool might be booked a lot, just a practical issue but a very real one. Also SEM costs hundreds of thousands of euros or millions. Nor cheap.

XRD is a method where you blast your sample with x-rays ans look at how they diffract from it. You can use it to probe which materials your sample is made of but you also get information about its crystal structure and things like distance between two atomic layers (very accurately). Issues are for instance that you might need to grind your sample into dust basically to do the measurement (it only probes the very surface of the sample). And it is not a high resolution method in the physical dimensions, you can tell what the entire sample is made of but not really what is this specific spot made of.

Then methods like Raman and infrared spectroscopy use lasers to excite molecules on the sample and then look at what the sample spews out. They both can be used to know what materials the sample is made of (at the laser spot), but not everything is "Raman active" or "infrared active". Like I mentioned, they probe molecules and not necessarily individual atoms. Essentially they look at how molecules vibrate and rotate and how the electron cloud around the atoms stretch and move when being hit by lasers. EDX might tell you a material is made of carbon 12, but how is it arranged? Amorphous carbon (no crystal structure)? Buckyballs (small clusters of couple tens of carbon atoms in a ball)? Carbon nanotubes (sheet of carbon rolled into a tube)? Graphene (2D sheet of carbon aroms)? Raman and/or IR spectroscopy can tell you that. Now to be fair, EDX can also differentiate between those (or the electron microscope can as a whole) but it will have though time telling how well the atoms are arranged (missing atoms, doping, extra atoms etc).

Of course you can just take white light (usually maybe 300-1000nm or so), shine the sample with it and look at the spectra. Either transmissed light (light that goes through) or reflected. Then you can run into issues like, well most stuff doesn't let white light through that easily for transmission, and not all samples reflect that well. Here you have looooots of different wavelengths so just making that wide frequency band well is difficult, hence it is usually limited to around the visible spectrum, and this also is a problem in spectral resolution (tends to be lower). And all frequencies interact with the sample a bit differently, so here afaik you don't really get any more info that literally what the sample reflects / passes through. So no crystal structure or anything fancy.

if elements drive the colors, how do you parse out individual elements from a compound?

You might not be able to. What you might see is how many percent of X and Y and Z you have and from this you could determine what type of a compound you have. You would probably start to look at phase diagrams of those elements and from there you might be able to determine what compound you have.

Then you could also run into issues with spectral resolution and non-idealistic measurement conditions. For instance you might see a peak of spectra at some wavelength, but the peak is not a single line, it probably looks like Gaussian curve or Lorentzian. Now you can have multiple peaks very close by but because of the resolution of the system isn't high enough to see them as individual peaks, you would see one big peak. To get around this you probably need to do some math and try to fit multiple peaks into your measurement data and see what peaks make up the big boi peak.

Is there a consistent pattern to how the emission lines relate to what the element looks like before going through the prism?

Pretty much yes. The theoretical peaks are what they are but your measurement data is noisy (like previous example). Your electrons are not at the same energy. You get secondary electrons that can mess up things. Your laser isn't at one frequency and might change a bit from measurement to measurement. If you need accurate results you might need to make some calibration steps. Like, measure something you know is 99.999% copper, then adjust the setup so that it identifies it correctly. For lasers you should measure the frequency and adjust the results based on that. You might also need to measure like spectrum of a xenon lamp that has very well known, strong peaks, and adjust according to that.

Is there a resource that shows examples of emission lines along with the visible color?

There are lots of books on spectroscopy of different materials that work as reference. NIST also has a database https://www.nist.gov/pml/atomic-spectra-database but I have not used it myself. But usually the tool itself has a database that you just query from and it tells you.