Young diagrams and $\mathrm{SU}(n)$ representations

Question

I'm trying to understand from a systematic point of view how $\mathrm{SU}(n)$ representations may be computed from Young diagrams following Sternberg's Group Theory and Physics. Chapter five gives means to calculate the dimension of each representation from the decomposition into irreducibles of the action of $S_r$ on $\otimes^r \mathbb{C}^n$. Nevertheless I don't see how one can actually compute the representation. If anyone has a good reference where an example is calculated I would appreciate it a lot!