Tensor product decompositions are well studied and there are algorithms at least in all classical cases. (By the way, it is a special kind of branching rule for diagonal inclusion $G \to G\times G.$) See e.g. Wikipedia page on LR rule and references therein. https://en.wikipedia.org/wiki/Littlewood%E2%80%93Richardson_rule

In general, it is a complicated problem with some remaining open questions. In practice, one has some particular constraints and an already chosen parametrisation (i.e. Young diagrams, highest weights, crystals, ...) and goes on from there. There is at least one case when the tensor product with fundamental representation has simple rule which goes under the name Pieri rule and that is the case of the standard representation $\mathbb{C}^n.$