We present early results from the COMAP Galactic Plane Survey conducted between June 2019 and April 2021, spanning $20^∘<\ell<40^∘$ in Galactic longitude and $|b|<1.\!\!^{∘}5$ in Galactic latitude with an angular resolution of $4.5^{\prime}$. The full survey will span $\ell ∼ 20^{∘}$- $220^{∘}$ and will be the first large-scale radio continuum survey at $30$ GHz with sub-degree resolution. We present initial results from the first part of the survey, including diffuse emission and spectral energy distributions (SEDs) of HII regions and supernova remnants. Using low and high frequency surveys to constrain free-free and thermal dust emission contributions, we find evidence of excess flux density at $30\,$GHz in six regions that we interpret as anomalous microwave emission. Furthermore we model UCHII contributions using data from the $5\,$GHz CORNISH catalogue and reject this as the cause of the $30\,$GHz excess. Six known supernova remnants (SNR) are detected at $30\,$GHz, and we measure spectral indices consistent with the literature or show evidence of steepening. The flux density of the SNR W44 at $30\,$GHz is consistent with a power-law extrapolation from lower frequencies with no indication of spectral steepening in contrast with recent results from the Sardinia Radio Telescope. We also extract five hydrogen radio recombination lines to map the warm ionized gas, which can be used to estimate electron temperatures or to constrain continuum free-free emission. The full COMAP Galactic plane survey, to be released in 2023/2024, will be an invaluable resource for Galactic astrophysics.