We report upper-limits on the Epoch of Reionization (EoR) 21 cm power spectrum at redshifts 7.9 and 10.4 with 18 nights of data ($\sim 36$ hours of integration) from Phase I of the Hydrogen Epoch of Reionization Array (HERA). The Phase I data show evidence for systematics that can be largely suppressed with systematic models down to a dynamic range of $\sim 109$ with respect to the peak foreground power. This yields a 95% confidence upper limit on the 21 cm power spectrum of $\Delta 212\le (30.76)2 mK2$ at $k=0.192 h Mpc-1$ at $z=7.9$, and also $\Delta 212\le (95.74)2 mK2$ at $k=0.256 h Mpc-1$ at $z=10.4$. At $z=7.9$, these limits are the most sensitive to-date by over an order of magnitude. While we find evidence for residual systematics at low line-of-sight Fourier $k\parallel$ modes, at high $k\parallel$ modes we find our data to be largely consistent with thermal noise, an indicator that the system could benefit from deeper integrations. The observed systematics could be due to radio frequency interference, cable sub-reflections, or residual instrumental cross-coupling, and warrant further study. This analysis emphasizes algorithms that have minimal inherent signal loss, although we do perform a careful accounting in a companion paper of the small forms of loss or bias associated with the pipeline. Overall, these results are a promising first step in the development of a tuned, instrument-specific analysis pipeline for HERA, particularly as Phase II construction is completed en route to reaching the full sensitivity of the experiment.