It is a pity you don't read French.  For those who do (or after you've learnt to read it), I would recommend *Annexe B* in the undergraduate text *Éléments d’analyse et d’algèbre (et de théorie des nombres)* by Pierre Colmez, and also his popular article [Un autre monde est possible][1].

**Addendum.** The basic idea for computing the volume of $\mathrm{SL}_2(\mathbf{A})/\mathrm{SL}_2(\mathbf{Q})$ is that this space can 
be identified with 
$$
(\mathrm{SL}_2(\mathbf{R})/\mathrm{SL}_2(\mathbf{Z}))
\times \mathrm{SL}_2(\mathbf{Z}_2)
\times \mathrm{SL}_2(\mathbf{Z}_3)
\times \mathrm{SL}_2(\mathbf{Z}_5)
\times\cdots
$$
so the volume in question is the product of the volumes of the various factors.  The computation of a *difficult* integral shows that the volume of $\mathrm{SL}_2(\mathbf{R})/\mathrm{SL}_2(\mathbf{Z})$ is $\zeta(2)$, and an *easy* computation shows that the volume of 
$\mathrm{SL}_2(\mathbf{Z}_p)$ is $1-{1\over p^2}$, for every prime $p$.  Finally, the eulerian product $\zeta(2)=\prod_p(1-{1\over p^2})^{-1}$ allows you to conclude that the volume of   $\mathrm{SL}_2(\mathbf{A})/\mathrm{SL}_2(\mathbf{Q})$ is $1$.

  [1]: http://images.math.cnrs.fr/Un-autre-monde-est-possible.html