You can add an edge label like this: \draw (t) -- node [anchor=south] {$p_1$} (q1). You will have to play around a bit with getting the anchors right (choose north, south west, etc., to see the effect). Other parameters you may want to look at is inner sep (which is the white space around the label) and pos (which determines where on the edge the label will appear).
Adding node weights can be done by \node[vertex, label=below:$10$](t) at (2, 0) {$ \omega_{i} $}; (also note that you don't need the extra []).
Here you go:
\documentclass{article}
\usepackage{tikz}
\begin{document}
\tikzstyle{vertex}=[circle, draw]
\begin{tikzpicture}[transform shape]
\node[vertex, label=right:$1$](t) at (4, 0) {$ \omega_{i} $};
\node[vertex, label=above:$2$](r1) at (2, 2) {$ r^{\omega_{i}}_{1} $};
\node[vertex, label=above:$3$](r2) at (6, 2) {$ r^{\omega_{i}}_{2} $};
\node[vertex, label=below:$4$](q1) at (0,-2) {$ q^{\omega_{i}}_{1} $};
\node[vertex, label=below:$5$](q2) at (4,-2) {$ q^{\omega_{i}}_{2} $};
\node[vertex, label=below:$6$](q3) at (8,-2) {$ q^{\omega_{i}}_{2} $};
\begin{scope}[every path/.style={-, dashed}, every node/.style={inner sep=1pt}]
\draw (t) -- node [anchor=north west] {$1$} (q1);
\draw (t) -- node [anchor=west] {$2$} (q2);
\draw (t) -- node [anchor=north east] {$3$} (q3);
\draw (t) -- node [anchor=north east] {$4$} (r1);
\draw (t) -- node [anchor=north west] {$5$} (r2);
\end{scope}
\begin{scope}[every path/.style={-}, every node/.style={inner sep=1pt}]
\draw (r1) -- node [anchor=east] {$6$} (q1);
\draw (r1) -- node [pos=0.15, anchor=north east] {$7$} (q2);
\draw (r1) -- node [pos=0.15, anchor=south west] {$8$} (q3);
\draw (r2) -- node [pos=0.15, anchor=south east] {$9$} (q1);
\draw (r2) -- node [pos=0.15, anchor=north west] {$10$} (q2);
\draw (r2) -- node [anchor=west] {$11$} (q3);
\end{scope}
\end{tikzpicture}
\end{document}

If you want your edge labels to be rotated automatically, you can use sloped:
\documentclass{article}
\usepackage{tikz}
\begin{document}
\tikzstyle{vertex}=[circle, draw]
\begin{tikzpicture}[transform shape]
\node[vertex, label=right:$1$](t) at (4, 0) {$ \omega_{i} $};
\node[vertex, label=above:$2$](r1) at (2, 2) {$ r^{\omega_{i}}_{1} $};
\node[vertex, label=above:$3$](r2) at (6, 2) {$ r^{\omega_{i}}_{2} $};
\node[vertex, label=below:$4$](q1) at (0,-2) {$ q^{\omega_{i}}_{1} $};
\node[vertex, label=below:$5$](q2) at (4,-2) {$ q^{\omega_{i}}_{2} $};
\node[vertex, label=below:$6$](q3) at (8,-2) {$ q^{\omega_{i}}_{2} $};
\begin{scope}[every path/.style={-, dashed}, every node/.style={sloped, inner sep=1pt}]
\draw (t) -- node [anchor=north] {$1$} (q1);
\draw (t) -- node [anchor=south] {$2$} (q2);
\draw (t) -- node [anchor=north] {$3$} (q3);
\draw (t) -- node [anchor=north] {$4$} (r1);
\draw (t) -- node [anchor=north] {$5$} (r2);
\end{scope}
\begin{scope}[every path/.style={-}, every node/.style={sloped, inner sep=1pt}]
\draw (r1) -- node [anchor=south] {$6$} (q1);
\draw (r1) -- node [pos=0.15, anchor=north] {$7$} (q2);
\draw (r1) -- node [pos=0.15, anchor=south] {$8$} (q3);
\draw (r2) -- node [pos=0.15, anchor=south] {$9$} (q1);
\draw (r2) -- node [pos=0.15, anchor=north] {$10$} (q2);
\draw (r2) -- node [anchor=south] {$11$} (q3);
\end{scope}
\end{tikzpicture}
\end{document}

\draw (t) -- node [anchor=south] {$p_1$} (q1);You will have to play around a bit with getting the anchors right (choosenorth,south west, etc to see the effect). Other parameters you may want to look at isinner sep(which is the white space around the label) andpos(which determines where on the edge the label will appear). – yori Dec 12 '14 at 09:00\node[vertex, label=below:$10$](t) at (2, 0) {$ \omega_{i} $};(also note that you don't need the extra[]). – yori Dec 12 '14 at 10:05