I am sure this is quite a basic question but I am having trouble understanding whole these key concepts are related
Say we had a field extension $L/K$ .. How do the concepts of being normal and separable relate to our understanding of the Galois group $Gal(L/K)$?
One of the characterizations of being a Galois extension (for a finite extension) is that it is normal and separable.
If you know that your extension is a Galois extension, then all sorts of wondrous things happen such as the order of the Galois group being the degree of the extension and many other nice things.
See here:
Fundamental Theorem of Galois Theory
Characterizations of Galois Extensions