ag.algebraic geometry - resolution of singularities on surfaces

Not necessarily, I don't think. There are surface singularities (though I can't recall an example easily, but one whose resolution has exceptional divisor an elliptic curve) for which if you blow up at a point you get a singular curve. The example I'm thinking of is in Kollar's Exercises.

In the case I'm thinking of, you have a surface with a single point singularity, you blow it up, and you get a rational curve singularity, which if you blow up (or normalize, either one) gives you an elliptic curve over it.

EDIT: found it, it's exercise 68, do all three parts to see some of the things that can happen.