I'm looking to do a similar calculation as in: Curvature tensor of 2-sphere using exterior differential forms (tetrads)
in terms of tetrads/Vierbeins etc but I can't find any literature on how one explicitly calculates the tetrads or even what they are. Any help or clarity would be greatly appreciated. Even if you could explain how to calculate this tetrad basis $e^\theta = rd\theta$ and $e^\phi = r \sin{\theta} d\phi$ in this case.
Any references on the Riemann curvature tensor in terms of tetrads and the torsion tensor would be greatly appreciated.
I believe this is covered extremely well in John Stewart's Advanced General Relativity. It is a short textbook but extremely rich in content, certainly on the tetrad representation moving to Newman-Penrose formalism.
Alternatively you could look in the introduction / Chapter 1 of Chandrasekhar's Mathematical Theory of Black Holes.