  We extend the results of Pareschi on the constancy of the gonality and
Clifford index of smooth curves in a complete linear system on Del Pezzo
surfaces of degrees $\geq 2$ to the case of Del Pezzo surfaces of degree 1,
where we explicitly classify the cases where the gonality and Clifford index
are not constant. We also classify all cases of exceptional curves on Del Pezzo
surfaces, which turn out to be the smooth plane curves and some other cases
with Clifford dimension 3. Moreover, the property of being exceptional holds
for all curves in the complete linear system.
  Furthermore, we relate the Clifford index and gonality of smooth curves in
$|L|$ to the higher order embedding properties of $|L+K_S|$. More precisely, we
show that for a nef line bundle $L$ on a Del Pezzo surface, $L+K_S$ is
birationally $k$-very ample if and only if all the smooth curves in $|L|$ have
gonality $\geq k+2$, and we also find numerical criteria for birational
$k$-very ampleness.
