This paper reports results for directed flow $v_{1}$ and elliptic flow $v_{2}$ of charged particles in Cu+Cu collisions at $\sqrt{s_{NN}}=$ 22.4 GeV at the Relativistic Heavy Ion Collider. The measurements are for the 0-60% most central collisions, using charged particles observed in the STAR detector. Our measurements extend to 22.4 GeV Cu+Cu collisions the prior observation that $v_1$ is independent of the system size at 62.4 and 200 GeV, and also extend the scaling of $v_1$ with $\eta/y_{\rm beam}$ to this system. The measured $v_2(p_T)$ in Cu+Cu collisions is similar for $\sqrt{s_{NN}} = 22.4-200$ GeV. We also report a comparison with results from transport model (UrQMD and AMPT) calculations. The model results do not agree quantitatively with the measured $v_1(\eta), v_2(p_T)$ and $v_2(\eta)$.