We present measurements of $E$-mode polarization and temperature-$E$-mode correlation in the cosmic microwave background (CMB) using data from the first season of observations with SPTpol, the polarization-sensitive receiver currently installed on the South Pole Telescope (SPT). The observations used in this work cover 100~\sqdeg\ of sky with arcminute resolution at $150\,$GHz. We report the $E$-mode angular auto-power spectrum ($EE$) and the temperature-$E$-mode angular cross-power spectrum ($TE$) over the multipole range $500 < \ell \leq5000$. These power spectra improve on previous measurements in the high-$\ell$ (small-scale) regime. We fit the combination of the SPTpol power spectra, data from \planck\, and previous SPT measurements with a six-parameter \LCDM cosmological model. We find that the best-fit parameters are consistent with previous results. The improvement in high-$\ell$ sensitivity over previous measurements leads to a significant improvement in the limit on polarized point-source power: after masking sources brighter than 50\,mJy in unpolarized flux at 150\,GHz, we find a 95\% confidence upper limit on unclustered point-source power in the $EE$ spectrum of $D_\ell = \ell (\ell+1) C_\ell / 2 \pi < 0.40 \ \mu{\mbox{K}}^2$ at $\ell=3000$, indicating that future $EE$ measurements will not be limited by power from unclustered point sources in the multipole range $\ell < 3600$, and possibly much higher in $\ell.$ Show Partial Abstract