I misinterpreted the question at first, sorry. Here's my new answer:

**First, an answer to the wrong question**

For two $(n-1)$ dimensional spheres of radii r and R with the metrics induced from embedding in $\mathbb{R}^n$ (note, this is the "chord metric" not the "round metric" as Zarathustra desired), the Gromov-Hausdorff distance is $|r-R|$. We can achieve this as an upper bound by embedding the two spheres in a concentric fashion, and it's seen to be sharp by the inequality $d_{GH}(X,Y)\geq \frac{1}{2}|diam(X)-diam(Y)|$.

See e.g. Burago Burago and Ivanov, ex. 7.3.14 which is a good source in general.

**Now an answer to the correct question**

The answer is $\frac{\pi}{2}|R-r|$ for spheres with the round metric, as Anton more or less suggested. This follows easily from the discussion following defn. 7.3.17 in BBI.

Out of laziness, I've written out some of the details here. A "correspondence" of metric spaces $X$ and $Y$ is defined to be a subset $\mathcal{R}$ of $X\times Y$ such that for every point $x\in X$ there is at least one point $(x,z)\in\mathcal{R}$ and for every $y\in Y$ there is at least one point $(w,y)\in\mathcal{R}$. From this one can prove Theorem 7.3.25 which states

$d_{GH}(X,Y)=\frac{1}{2}\inf_{\mathcal{R}}dis\mathcal{R}$

where the infimum is taken over all correspondences $\mathcal{R}$ and $dis\mathcal{R}$ is the distortion of $\mathcal{R}$, defined to be $\sup\{|d_X(x,x')-d_Y(y,y')|:(x,y),(x',y')\in\mathcal{R}\}$.

Take $\mathcal{R}$ to be the correspondence consisting of pairs $(x,y)$ with $x\in S^2_{r}$ and $y\in S^2_{R}$ if $x$ and $y$ lie on the same ray through the origin when the two spheres are embedded in $\mathbb{R}^3$. The distortion of this correspondence is $\pi|R-r|$ by taking $x$ and $x'$ to be antipodal points on one of the spheres. This gives an upper bound for $d_{GH}$ of $\frac{\pi}{2}|R-r|$, and this is sharp again by the inequality above.

**A possibly useful reference for G-H distance for subsets of Euclidean space in general**

For the rest of your question, you might find interesting a paper by Facundo Mémoli which discusses the case when X and Y are subsets of Euclidean space. See also slides here.