How to add newline within node using TikZ?

One simple method is to specify text characteristics within the node : text width, etc. This will let you do exactly what you want, without any extra package. For example,

\documentclass[]{minimal}

\usepackage{amsmath,amsfonts,amssymb}
\usepackage{tikz}
\usetikzlibrary{automata,positioning}

\begin{document}

\begin{tikzpicture}[shorten >=1pt,node distance=5cm,on grid,auto] 

\node[state,initial] (q_0) {$q_0$}; 
\path[->] (q_0) edge[loop above] node[text width=1cm,align=center] {0,1,2\\3,4,5} (q_0); 

\end{tikzpicture}
\end{document}  

The result is

Based on Frédérics answer and Chans comment you can just do:

\documentclass[]{minimal}

\usepackage{amsmath,amsfonts,amssymb}
\usepackage{tikz}
\usetikzlibrary{automata,positioning}

\begin{document}

\begin{tikzpicture}[shorten >=1pt,node distance=5cm,on grid,auto] 

\node[state,initial] (q_0) {$q_0$}; 
\path[->] (q_0) edge[loop above] node[align=center] {0,1,2\\3,4,5} (q_0); 

\end{tikzpicture}
\end{document}  

You don't need the text width option, just the align

You could use the makecell package. As stated in the package documentation, it provides

\makecell[<vertical or/and horizontal alignment>]{<cell text>}

that aids in the creation of (small-scale) multi-lined tabular cell. In that regard, consider the following alteration to your code:

...
\usepackage{makecell}%
...
\path[->]
  (q_0) edge node {0,1} (q_1)
        edge [loop above] node {\makecell[l]{0,1,2,\\3,4,5}} (q_0)
  (q_1) edge node {0,1} (q_2)
  (q_2) edge [loop right] node {1} (q_2)
 ; %end path