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The layered perovskites $R$Cr(BO$_3$)$_2$ ($R$ = Y and Ho) with magnetic triangular lattices were studied by performing ac/dc susceptibility, specific heat, elastic and inelastic neutron scattering, and dielectric constant measurements. The results show (i) both samples' Cr$^{3+}$ spins order in a canted antiferromagnetic structure with $T_N$ around 8-9 K, while the Ho$^{3+}$ ions do not order down to $T$ = 1.5 K in HoCr(BO$_3$)$_2$; (ii) when a critical magnetic field H$_{C}$ around 2-3 T is applied below $T_{N}$, the Cr$^{3+}$ spins in the Y-compound and both the Cr$^{3+}$ and Ho$^{3+}$ spins in the Ho-compound order in a ferromagnetic state; (iii) both samples exhibit dielectric constant anomalies around the transition temperature and critical field, but the Ho-compound displays a much stronger magnetodielectric response. We speculate that this is due to the magnetostriction which depends on both of the Cr$^{3+}$ and the Ho$^{3+}$ ions' ordering in the Ho-compound. Moreover, by using linear spin wave theory to simulate the inelastic neutron scattering data, we estimated the Y-compound's intralayer and interlayer exchange strengths as ferromagnetic J$_{1}$ = -0.12 meV and antiferromagnetic J$_{2}$ = 0.014 meV, respectively. The competition between different kinds of superexchange interactions results in the ferromagnetic intralayer interaction.