Flavor physics measurements at high luminosity $B$-factories offer a good probe for testing the Standard Model and looking for New Physics. With the first successful $e^+e^-$ collisions recorded in 2018, the Belle II experiment is accumulating its first physics data. The design peak instantaneous luminosity of the SuperKEKB accelerator is $6\times 10^{35}~\mathrm{cm}^{-2}\mathrm{s}^{-1}$. The size of the targeted data set is 50~ab$^{-1}$, which will significantly improve the experimental precision on the three angles of the CKM unitarity triangle measured by the first generation of $B$ factories.. These measurements are based on time-dependent $CP$ asymmetry analyses for the angles $\phi_1$ and $\phi_2$, and on the measurement of direct $CP$ violation in the decay channel $B^- \to DK^-$ for $\phi_3$. In this proceeding, we report the prospects for determining $\phi_3$. In addition, we describe the first calibration and performance of the Belle II flavor tagging algorithm. Finally, we present the first time-dependant $CP$ violation measurement using the channel $B^0 \to J/\psi (e^-e^+/ \mu^- \mu^+) K_S$ with the 34.6 fb$^{-1}$ data set recorded by Belle II so far, which results in the measurement of $\sin2\phi_1$ of $0.55 \pm 0.21 \pm 0.04$ where the first and second uncertainties are statistical and systematic, respectively.