I will soon (self)study the topic of integration in depth, for which I'm planning to use Rudin. However, Rudin seems to treat the Riemann-Stieltjes integral, which seems more advanced and general than the Riemann integral (and which also gives the Riemann integral (?)).
Will I miss something if I immediately do Riemann-Stieltjes integration, or should I do ordinary Riemann-integral first?
Let me tell your my experience as a student. My instructor chose Rudin's book for our course, and he taught directly integration theory according to Riemann-Stieltjes. Honestly, had I not studied the fundamentals of the classical Riemann integral at high school, I would have not understood the whole meaning of the R.-S. integral.
It is surely a powerful tool, but it might be hard to compute (except in the easy case of a smooth weight function) and seldom used in applications. Even in Rudin's book, the most important results hold true for the Riemann integral.
My advice is to properly understand the Riemann integral ($\alpha \equiv 1$ in Rudin's notation). When you feel at ease with it, try to move on and consider the R.-S. integral.