Flow harmonics ($v_n$) in the Fourier expansion of the azimuthal distribution of particles are widely used to quantify the anisotropy in particle emission in high-energy heavy-ion collisions. The symmetric cumulants, SC($m,n$), are used to measure the correlations between different orders of flow harmonics. These correlations are used to constrain the initial conditions and the transport properties of the medium in theoretical models. In this Letter, we present the first measurements of the four-particle symmetric cumulants in Au+Au collisions at $\sqrt{s_{NN}}$ = 39 and 200 GeV from data collected by the STAR experiment at RHIC. We observe that $v_2$ and $v_3$ are anti-correlated in all centrality intervals with similar correlation strengths from 39 GeV Au+Au to 2.76 TeV Pb+Pb (measured by the ALICE experiment). The $v_2$-$v_4$ correlation seems to be stronger at 39 GeV than at higher collision energies. The initial-stage anti-correlations between second and third order eccentricities are sufficient to describe the measured correlations between $v_2$ and $v_3$. The best description of $v_2$-$v_4$ correlations at $\sqrt{s_{NN}}$ = 200 GeV is obtained with inclusion of the system's nonlinear response to initial eccentricities accompanied by the viscous effect with $η/s$ > 0.08. Theoretical calculations using different initial conditions, equations of state and viscous coefficients need to be further explored to extract $η/s$ of the medium created at RHIC.