Why does DFS only yield tree and back edges on undirected, connected graphs?
A depth first search on a directed graph can yield 4 types of edges; tree, forward, back and cross edges. As we are looking at undirected graphs, it should be obvious that forward and back edges are the same thing, so the only things left to deal with are cross edges.

A cross edge in a graph is an edge that goes from a vertex $v$ to another vertex $u$ such that $u$ is neither an ancestor nor descendant of $v$. So what you need to argue is that in an undirected graph, there's no way you can get a cross edge. It might help to think of why the can occur in directed graphs, and why you can't have this case in undirected graphs.