Using proton-proton collision data, corresponding to an integrated luminosity of $9fb-1$ collected with the LHCb detector, seven decay modes of the $Bc+$ meson into a $J/\psi$ or $\psi (2S)$ meson and three charged hadrons, kaons or pions, are studied. The decays $B_c^+\rightarrow ( \psi(2S) \rightarrow J/\psi \pi^+ \pi^-)\pi^+ $, $Bc+\to \psi (2S)\pi +\pi +\pi -$, $Bc+\to J/\psi K+\pi +\pi -$ and $Bc+\to K+K+K-$ are observed for the first time, and evidence for the $Bc+\to \psi (2S)K+K-\pi +$ decay is found, where $J/\psi$ and $\psi (2S)$ mesons are reconstructed in their dimuon decay modes. The ratios of branching fractions between the different $Bc+$ decays are reported as well as the fractions of the decays proceeding via intermediate resonances. The results largely support the factorisation approach used for a theoretical description of the studied decays.