I would like to ask if there exist pedagogical expositions of the MordellWeil theorem (wikipedia). What parts of number theory (algebraic geometry) one should better learn first before starting to read a proof of MordellWeil?

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 MordellWeil, NagellLutz Theorem, elliptic curves over finite fields, etc. For more advanced treatment of MordellWeil, I suggest the following textbook: J. Silverman "The arithmetic of elliptic curves" (Chapter 8 is about MordellWeil). 


For the case of elliptic curves, there is Mordell's proof, discussed in his book Diophantine Equations (pp. 138148). I could hardly imagine less prerequisites than this. 


There is a very elementary and selfcontained (modulo a few things proved earlier in the book) proof in Chapter 19 of the book of Ireland and Rosen, "A classical introduction to modern number theory". One might object that it can be misleading to use explicit but obscure polynomial identities instead of more intrinsic facts from algebraic geometry, but the text has lots of good remarks and references to go beyond this elementary approach. 


Manin's proof of MordellWeil theorem (for abelian varieties over number fields) has appeared as an appendix to Russian translation of First edition of Mumford's ``Abelian varieties". Eventually it was translated into English and published as an appendix to Second and Third editions of Mumford's book. 


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


Actually, the wikipedia article you cite cites Joe Silverman's book, which contains such a "pedagogical" exposition. The book is not entirely selfcontained, but I am sure the preface explains the prerequisites. 


I think one should also mention Jean Pierre Serre Lectures on the MordellWeil Theorem Aspects of Mathematics 


Already mentioned: Silverman and Tate's "Rational Points on Elliptic Curves" (undergraduate level) and Silverman's "The Arithmetic of Elliptic Curves" (graduate level). Another text at the undergraduate level that covers Mordell's theorem (i.e., the MordellWeil theorem for elliptic curves over $\mathbb{Q}$) is Washington's "Elliptic Curves: Number Theory and Cryptography" (see Chapter 8). If you are looking for a proof of the MordellWeil theorem in its utmost generality (i.e., for abelian varieties over number fields), I would suggest Hindry and Silverman's "Diophantine Geometry: An Introduction" (see Part C). 

