In order to talk about the volume you need to fix the measure first. Weil observed that there is a canonical choice for the Haar measure on $G(\mathbb{A})$. For a connected $n$-dimensional semisimple algebraic group $G$ over a global field $K$, there is a nonzero $n$-dimensional rational differential $K$-form which is invariant under left-translation. The differential form induces a Haar measure on $G(K_v)$ for each completion $K_v$ of $K$, hence it determines a Haar measure on the restricted direct product $G(\mathbb{A})$ as well. The differential form is unique up to scaling by $K^\times$, hence the obtained Haar measure on $G(\mathbb{A})$ is unique. When $G$ is simply connected and semisimple, Weil conjectured that $\mathrm{vol}(G(\mathbb{A})/G(K))=1$ for this particular measure.

In the case of $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$, the invariant differential form is given by $\frac{dx\wedge dy\wedge dz}{x}$, where $\begin{pmatrix} x & y \\ z & t \end{pmatrix}$ are the usual coordinates on the group. For this form one obtains the numbers mentioned by Chandan Singh Dalawat (and also by Colmez's [Un autre monde est possible][1]), hence a proof of Weil's conjecture for $G=\mathrm{SL}_2$ and $K=\mathbb{Q}$. For details I recommend Platonov-Rapinchuk: Algebraic groups and number theory (Academic Press, 1994). See especially Example 3 on pp. 166-167, the Example on pp. 222-223, and the Example on p. 262.

**Added.** The OP is looking for a direct (global) proof of $\mathrm{vol}(\mathrm{SL}_2(\mathbb{A})/\mathrm{SL}_2(\mathbb{R}))=1$. For the analogous (local) statement that $\mathrm{vol}(\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z}))=\zeta(2)$ there are nice accounts via Eisenstein series, see for example [here][2]. No doubt this proof can be modified to yield $\mathrm{vol}(\mathrm{SL}_2(\mathbb{A})/\mathrm{SL}_2(\mathbb{Q}))=1$ directly as well, but I don't know of any simple written account. Hopefully someone knows a good reference. In general, Langlands proved Weil's conjecture for simply connected Chevalley groups with the help of his general theory of Eisenstein series. 


  [1]: http://images.math.cnrs.fr/Un-autre-monde-est-possible.html
  [2]: http://www.math.umn.edu/~garrett/m/v/volumes.pdf