I assume $S$ is a projective smooth toric surface.

If $D_1, \dots, D_n$ are the irreducible toric divisors on $S$, then $-K_S=D_1+\dots+D_n$ is an anticanonical divisor. Thus blowing up any point on any of these divisors one obtains a smooth anticanonical rational surface $\tilde S$; such surfaces are very well known by work of Brian Harbourne. Maybe more simply, by Mori theory, since $-K_{\tilde S}=-\pi^*K_S-E$ is effective on $\tilde S$, the cone of curves is spanned by extremal rays $\mathbb{R}C_i$ with $-K_{\tilde S}\cdot C_i>0$ and components of $-K$; *these curves are in our case the exceptional divisor and the birational transforms of a subset of* $D_i$'s [**EDIT**: the previous (italicized) sentence was not correct because new extremal rays do appear with the blowup, but it is still true that these extremal rays can be controlled with Harbourne's results]. Thus to determine nefness on $\tilde S$ and so the Seshadri constant of any ample divisor at the given point is not difficult. In particular these Seshadri constants are rational, and they only depend on the ample class and the $D_i$(s) to which the point belongs. [**EDIT**: these conclusions are still correct].

Blowing up at a general point of $S$ may give a surface which is anticanonical (if $-K$ is not fixed on $S$) or non-anticanonical (when $|-K|=\{-K\}$). In the first case, similar considerations would lead to the computation of the Seshadri constant. In the second case, I am afraid the problem can be difficult, and indeed related to the Nagata conjecture. The simplest interesting example would be the following: start with $\mathbb{P}^2$ as a toric surface and blow it up at the three three toric points. Now blow up the resulting surface at its six toric points. The result is a toric surface, the blow up of $\mathbb{P}^2$ at three clusters of three infinitely near points, where the Seshadri constant of your preferred ample divisor $L$ at a *general* point is presumably unknown. It might be irrational, if $L^2$ is not a square.

An analogon of the Nagata conjecture for toric surfaces (Seshadri constants at sets of $r\gg 0$ general points) can of course be stated, as particular cases of the conjecture stated by Lazarsfeld in 5.1.24 of "Positivity in Algebraic Geometry". I have nothing particularly relevant to say about that, except that it will probably be just as difficult as Nagata's original conjecture.