You need to learn about Clebsch-Gordan series and Young diagrams (see here) or the application of the Littlewood-Richardson rule for $su(n)$ decomposition. This is standard fare and there are plenty of textbooks around, for instance Lichtenberg's Unitary Symmetry and Elemenatry particles. There is a review by Dick Slansky that contains lots of useful stuff, but not so much on decomposing irreps. Have fun.
The "game" can be applied in modified form to the representations of the other classical series, but the rules loose their simplicity.
If you are looking for perfectly general algorithms that will work for the classical series, then you need to look at Schur functions but this is not so easy. Basically Schur functions are characters, and one decomposes a product by expanding in Schur functions the product of two Schur functions. This is not practical to do by hand for anything but the simplest cases. Bryan Wybourne did a lot to introduce Schur functions in physics. There is also a lot of research-grade work by Ron King and his students.
If you are primarily interested in the computational aspect, there are now packages, such as LieART that work quite well. LieArt is by no means the only one and I am not endorsing this one over others; it is pretty easy to read the documentation. See also this and this from mathoverflow if you need other suggestions.