In this paper, we give a characterization of Fredholmness of the Toeplitz operators on the Bergman spaces
A
φ
p
with exponential weights in
D
when
0
<
p
<
∞
. Also, we obtain the sufficient and necessary conditions which the Toeplitz and Hankel operators on
A
φ
2
belong to
h
-Schatten class, where
h
is a continuous increasing convex function.