  We derive cosmological constraints using a galaxy cluster sample selected
from the 2500~deg$^2$ SPT-SZ survey. The sample spans the redshift range $0.25<
z<1.75$ and contains 343 clusters with SZ detection significance $\xi>5$. The
sample is supplemented with optical weak gravitational lensing measurements of
32 clusters with $0.29<z<1.13$ (from Magellan and HST) and X-ray measurements
of 89 clusters with $0.25<z<1.75$ (from Chandra). We rely on minimal modeling
assumptions: i) weak lensing provides an accurate means of measuring halo
masses, ii) the mean SZ and X-ray observables are related to the true halo mass
through power-law relations in mass and dimensionless Hubble parameter $E(z)$
with a-priori unknown parameters, iii) there is (correlated, lognormal)
intrinsic scatter and measurement noise relating these observables to their
mean relations. We simultaneously fit for these astrophysical modeling
parameters and for cosmology. Assuming a flat $\nu\Lambda$CDM model, in which
the sum of neutrino masses is a free parameter, we measure
$\Omega_\mathrm{m}=0.276\pm0.047$, $\sigma_8=0.781\pm0.037$, and
$\sigma_8(\Omega_\mathrm{m}/0.3)^{0.2}=0.766\pm0.025$. The redshift evolution
of the X-ray $Y_\mathrm{X}$-mass and $M_\mathrm{gas}$-mass relations are both
consistent with self-similar evolution to within $1\sigma$. The mass-slope of
the $Y_\mathrm{X}$-mass relation shows a $2.3\sigma$ deviation from
self-similarity. Similarly, the mass-slope of the $M_\mathrm{gas}$-mass
relation is steeper than self-similarity at the $2.5\sigma$ level. In a $\nu
w$CDM cosmology, we measure the dark energy equation of state parameter
$w=-1.55\pm0.41$ from the cluster data. We perform a measurement of the growth
of structure since redshift $z\sim1.7$ and find no evidence for tension with
the prediction from General Relativity. We provide updated redshift and mass
estimates for the SPT sample. (abridged)
