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In this paper, we focus on the weighted Bergman spaces $A_{\varphi}^{p}$ in$\mathbb{D}$ with $\varphi\in\mathcal{W}_{0}$. We first give characterizationsof those finite positive Borel measures $\mu$ in $\mathbb{D}$ such that theembedding $A_{\varphi}^{p}\subset L_{\mu}^{q}$ is bounded or compact for$0<p,q<\infty$. Then we describe bounded or compact Toeplitz operators$T_{\mu}$ from one Bergman space $A_{\varphi}^{p}$ to another $A_{\varphi}^{q}$for all possible $0<p,q<\infty$. Finally, we characterize Schatten classToeplitz operators on $A_{\varphi}^{2}$.

Toeplitz operators on Bergman spaces with exponential weights