I am preparing a talk on the Eightfold Way, and am attempting to explain the spectra of the light mesons/baryons via representation theory. It will be delivered to students who have never seen representation theory before. I understand the arrangement of particles can be explained by reps of $\mathrm{SU}(3)$, e.g. the light mesons ($\mathrm{u}$, $\mathrm{d}$, $\mathrm{s}$) correspond to the representation $\mathbf{3} \otimes \mathbf{\bar{3}}$ of $\mathrm{SU}(3)$, which I understand decomposes as $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$.
Is there a simple and elementary way to derive this decomposition, i.e. without needing to know a lot of Lie groups/rep theory?
I've seen Young diagrams used as a tool, but still have not been able to understand how they work. If someone could give a self-contained explanation, or direct me to where I could find one, that would be great. I know group theory and surface-level stuff, and am willing to blindly accept some facts [e.g. that there exist representations $\mathbf{1}$, $\mathbf{3}$, $\mathbf{6}$, $\mathbf{8}$ of $\mathrm{SU}(3)$], but I'd like to give some motivation for the formula $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$ without just handwaving it.
Same for the baryons: $\mathbf{3} \otimes \mathbf{3} \otimes \mathbf{3} = \mathbf{10} \oplus \mathbf{8} \oplus \mathbf{8} \oplus \mathbf{1}$.
