@usai you might not find the responses above completely helpful? :) :)
Seriously, kidding aside, PhysicsForums is generally better at helping people over the hard spots than laying out an entire lecture-equivalent tutorial (better done by textbooks and online course materials). So if you can get hold of a good online explanation of the algebra of the spin operators - several out there and some other poster may chime in with their favorite - and start in on it, come back with a more focused question as soon as you hit a hard spot you'll get better and more helpful answers.
Another way of thinking about it: Suppose you are taking a college-level intro to QM course as part of a physics bachelor's degree (which is about right for this topic and consistent with the I rating you gave the thread). You walk into your professor/TA's office during office hours with this question, the absolute first thing you're going to hear is "How far into the textbook/course notes did you get?". Same thing here, except with the general unstructuredness of the internet layered on top.