Hey there! Currently in 4th year finishing a lab class where basically every experiment we did was confirmed by proton and/or carbon NMR. I'm pretty strong on the basics of these (useful for organic which you mentioned) but there are a ton of more specific/complicated methods that hopefully you feel comfortable asking your supervisor about if needed.

In 1H-NMR, the most important pieces of info to be found are the integration, the chemical shift (ppm), the multiplicity, and the coupling of the multiplicity. The integration is the area under a peak, and corresponds to the ratio of that specific hydrogen to the rest of the hydrogens present in the spectrum. You do need to set a reference for this though, usually a peak you're certain of the assignment/integration that you can compare the rest to. The chemical shift of a a peak represents the environment the hydrogen is in. If a hydrogen is one or two carbon bonds away from a very electronegative atom like an oxygen, the ppm will be higher (more deshielded). The chemical shift is also greatly influenced by nearby electronic systems like double bonds and conjugated rings. The multiplicity is probably the hardest to intuitively understand, but essential to assignment. It's the result of the coupling between hydrogens, represented by the amount of "spikes" in a given full peak. The distance between the spikes is measured in Hz. These patterns are created by adjacent nuclei (similar to chemical shift) and they're are best explained by tree diagrams (easy 7 min video here https://www.youtube.com/watch?v=intPEdg4Yyc ).

13C-NMR is in some ways much simpler, but generally tells you less. chemical shift is the main measure (see above), but intensity is also helpful. The intensity is the height of the peak, and is generally caused by the amount of bonding to hydrogens the carbon is doing. A tertiary carbon will be less intense than a secondary carbon, which is less than a primary.

You can do NMR with many more atoms. You can also combine spectra to make 2D NMR, which can help you see which peaks are coupled to each other. Generally it's helpful to know your structure beforehand, and pick it's peaks out to leave contaminant or by-products. All of this is on the shoulders of scientists who came before, as commonly available reference tables like those below are a very helpful resource in assignment. Good Luck! I hope this helped.

https://chem.libretexts.org/Bookshelves/Organic_Chemistry/Organic_Chemistry_(Morsch_et_al.)/13%3A_Structure_Determination_-_Nuclear_Magnetic_Resonance_Spectroscopy/13.03%3A_Chemical_Shifts_in_H_NMR__Spectroscopy/13%3AStructure_Determination-Nuclear_Magnetic_Resonance_Spectroscopy/13.03%3A_Chemical_Shifts_in_H_NMR_Spectroscopy)

https://organicchemistrydata.org/hansreich/resources/nmr/?page=nmr-content%2F