  Given a regular weight $\omega$ and a positive Borel measure $\mu$ on the
unit disc $\mathbb{D}$, the Toeplitz operator associated with $\mu$ is
  $$
  \mathcal{T}_\mu(f)(z)=\int_{\mathbb{D}}
f(\zeta)\bar{B_z^\omega(\zeta)}\,d\mu(\zeta),
  $$ where $B^\omega_{z}$ are the reproducing kernels of the weighted Bergman
space $A^2_\omega$. We describe bounded and compact Toeplitz operators
$\mathcal{T}_\mu:A^p_\omega\to A^q_\omega$, $1<q,p<\infty$, in terms of
Carleson measures and the Berezin transform
  $$
  \widetilde{\mathcal{T}_\mu}(z)=\frac{\langle\mathcal{T}_\mu(B^\omega_{z}),
B^\omega_{z} \rangle_{A^2_\omega}}{\|B_z^\omega\|^2_{A^2_\omega}}.
  $$ We also characterize Schatten class Toeplitz operators in terms of the
Berezin transform and apply this result to study Schatten class composition
operators.
