Weil's Basic Number Theory is not "basic" in the sense of elementary, and I wouldn't recommend starting there. Since you're doing independent study, you might want a more gentle introduction. Stewart and Tall have a nice introduction to algebraic number theory. Ireland and Rosen's Classical Introduction to Modern Number Theory is also worth looking at. For a readable problem-based introduction to algebraic number theory, look at Marcus's Number Fields.

Hartshorne is a standard introduction, but I prefer the three-volume series by Ueno, published by the AMS in their Translations of Mathematical Monographs series.