Current versions of Mathematica enable an even more direct solution than Craig's, although possibly not as instructive
In[12]:= PDF[
TransformedDistribution[
x + y, {x \[Distributed] NormalDistribution[\[Mu], \[Sigma]],
y \[Distributed] ExponentialDistribution[\[Lambda]]}]][z]
Out[12]= 1/2 E^(
1/2 \[Lambda] (-2 z +
2 \[Mu] + \[Lambda] \[Sigma]^2)) \[Lambda] Erfc[(-z + \[Mu] + \
\[Lambda] \[Sigma]^2)/(Sqrt[2] \[Sigma])]
