After a good look for anything in the way of an explicit example, and harassing the algebraic geometers in the department, I still am unable to find an answer to my question, so any light will be very much appreciated.

Suppose we have blown up $\mathbb{P}^2$ at some singular points (of a mapping), and we find that in this new space we have 3 curves which have self-intersection -1, which we would like to blow down. In my particular case I have two degree 1 curves (which pass through 2 base points in $\mathbb{P}^2$) and a degree 2 curve which passes through 5 base points in $\mathbb{P}^2$. Given that I have these curves explicitly, is it possible to find the blow down of each curve, and if so how?

Also, what happens to the Picard group in this blown down space? If one of my curves is $H-E_1-E_2$ in my blown up space, where $H$ is the representative of a generic degree 1 curve in $\mathbb{P}^2$ and $E_1$ and $E_2$ are the total transforms of two base points, then what is the Picard group in this space where we have blown down the line $H-E_1-E_2$?

My experience has been that all these things are theoretically calculable but nobody seems to apply the theory in specific cases.

Many thanks!