Proofs of Mordell-Weil theorem

J. Silverman and J. Tate "The rational points on elliptic curves" is a wonderful introduction to elliptic curves over rational numbers. It covers topics such as Mordell-Weil, Nagell-Lutz Theorem, elliptic curves over finite fields, etc.

For more advanced treatment of Mordell-Weil, I suggest the following textbook:

J. Silverman "The arithmetic of elliptic curves" (Chapter 8 is about Mordell-Weil).

For the case of elliptic curves, there is Mordell's proof, discussed in his book Diophantine Equations (pp. 138-148). I could hardly imagine less prerequisites than this.

There must be a proof in Cassels' Lectures on elliptic curves (Cambridge University Press, Cambridge, 1991).

See also his masterly survey Diophantine equations with special reference to elliptic curves (J. London Math. Soc. 41 (1966) 193–291) and the historical essay Mordell's finite basis theorem revisited (Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 1, 31–41).

Here is a quote from this last paper :

Weil's generalization of Mordell's theorem (and subsequent generalizations) was usually referred to as the Mordell-Weil Theorem. Mordell himself strongly disapproved of this usage and frequently insisted (in public and in private) that what he had proved should be called Mordell's Theorem and that everything else could, for his part, be called simply Weil's Theorem.

Addendum. Another excellent source is Knapp's Elliptic curves (Princeton University Press, Princeton, 1992) which contains a proof of Mordell's theorem (over $\mathbf Q$).

There is a very affordable book by Milne (Elliptic curves, BookSurge Publishers, Charleston, 2006) and a very motivating one by Koblitz (Introduction to elliptic curves and modular forms, Springer, New York, 1993). Tate's Haverford Lectures also served as the basis for Husemoller (Elliptic curves, Springer, New York, 2004).