Thanks to everyone who answered! A (CW'ed) summary of some of what I learned:

In the first place, I now cheerfully second Greg Martin's recommendation of Chapter 5.1 of Montgomery and Vaughan. It is a rather "lowbrow", very readable treatment. (doesn't prove Mellin inversion in complete generality)

Also, as Matt Young pointed out, for any complex $s$, the function $t \rightarrow t^s$ is a character on $\mathbb{R}^{\times}$. This is a triviality, but the importance of this fact escaped me the first time. The invariant measure on $\mathbb{R}^{\times}$ is $\frac{dx}{x}$, and so the Fourier transform of a function $f$ defined on this group is exactly

$$\int_{x \in \mathbb{R}^{\times}} f(x) x^s \frac{dx}{x},$$

the Mellin transform. Once this is written down, the rest follows mechanically (from change of variables and Fourier inversion).

Thanks to all!