How can I represent a set of graphs for learning purposes?
I don't think that paper is suitable for what you want to do.  That paper generates an embedding of nodes: given a graph $G$ and a node $v$ in $G$, it outputs an embedding of $v$ (a representation of some properties of $v$).  You want an embedding of the entire graph: given a graph $G$, you want to compute an embedding of $G$ (a representation of the entire graph $G$).

So, I suspect you'll need to consider a different approach.  A usual starting point would be to do exactly what @Juho recommended: brainstorm some features that might be relevant (e.g., number of nodes, average degree), then map the graph to a feature vector (a list of the values of these features), and use the feature vector as your representation of the graph.  More sophisticated methods are possible as well, but I'd recommend you start with the simple approach before diving into more complex methods, especially if you are not very experienced in this field.