Lepton flavour universality as predicted by the Standard Model is challenged by various measurements of $B$-meson decays involving tau and charged light leptons. The Belle II experiment started recording first collision data in spring 2018 and offers a unique laboratory to study lepton flavour universality in leptonic and semileptonic $B$ decays at high precision. Here we report the status of establishing first measurements of (semi) tauonic $B$ decays and preliminary results of the $B\to\tau\bar{\nu}_{\tau}$ channel using the available Belle II data, and discuss the future potential of investigating $R(D^{(*)})$ anomalies.