Many number theorists, including me, learned Galois cohomology first via the proof of the Mordell-Weil theorem. The last chapter of Joe Silverman's book The Arithmetic of Elliptic Curves is a good source for this. It's very concrete and when you understand the proof you'll understand a lot about why number theorists like the cohomological formalism.

Many number theorists, including me, learned Galois cohomology first via the proof of the Mordell-Weil theorem. The last chapter of Joe Silverman's book The Arithmetic of Elliptic Curves is a good source for this. It's very concrete and when you understand the proof you'll understand a lot about why number theorists like the cohomological formalism.

Edit: But note that you'll learn just about H^1, not anything higher. It's a start!