The scattering of orthopositronium from He, Ne, Ar, ${\mathrm{Li}}^{+},$ ${\mathrm{Na}}^{+},$ and ${\mathrm{K}}^{+}$ is investigated with the fixed-core stochastic variational method. The scattering length for $\mathrm{Ps}\ensuremath{-}\mathrm{He}$ scattering was ${1.57a}_{0},$ a value consistent with the threshold cross sections derived from three positron lifetime experiments. The scattering lengths for the $\mathrm{P}\mathrm{s}\ensuremath{-}\mathrm{N}\mathrm{e}$ and $\mathrm{P}\mathrm{s}\ensuremath{-}\mathrm{A}\mathrm{r}$ systems were ${1.55a}_{0}$ and ${1.79a}_{0},$ respectively. That there was only a 15% variation among the scattering lengths is compatible with the experiment of Coleman et al. [J. Phys. B 27, 981 (1994)], who found the low-energy cross sections for He, Ne, and Ar to be about the same size. The scattering lengths for Ps scattering from ${\mathrm{Li}}^{+},$ ${\mathrm{Na}}^{+},$ and ${\mathrm{K}}^{+}$ were ${12.9a}_{0},$ ${28.5a}_{0},$ and $\ensuremath{-}{1.9a}_{0}.$ The relatively small magnitude of the ${\mathrm{P}\mathrm{s}\ensuremath{-}\mathrm{K}}^{+}$ scattering length strongly supports previous suggestions that the ${\mathrm{P}\mathrm{s}\ensuremath{-}\mathrm{K}}^{+}$ system does not support a bound state. The annihilation parameter ${}^{1}{Z}_{\mathrm{eff}}$ has also been computed as part of the analysis. The present values of ${}^{1}{Z}_{\mathrm{eff}}$ are about 2.5--3 times smaller than the accepted experimental values since short-range electron-positron correlations were not taken into consideration when the annihilation matrix element was evaluated.