Breaking equations within the align environment

Two more alternatives:

\documentclass[border=3mm,preview]{standalone}
\usepackage{mathtools}

    \begin{document}
    The first option:
\begin{align*}
[((a_0, a_1, a_2) + & (b_0, b_1, b_2)) + (c_0, c_1, c_2)]         \\
    & = [((a_0 + b_0, a_1 + b_1, a_2 + b_2)) + (c_0, c_1, c_2)] \\
    & = (  a_0 + b_0 + c_0, a_1 + b_1 + c_1, a_2 + b_2 + c_2)  
\end{align*}
The second one:
\begin{align*}
[((a_0, a_1, a_2) + (b_0, b_1, b_2)) + (c_0, c_1, c_2)]     
    & = \begin{multlined}[t]
            [((a_0 + b_0, a_1 + b_1, a_2 + b_2))\\
                 + (c_0, c_1, c_2)] 
        \end{multlined}\\
    & = \begin{multlined}[t]
            (  a_0 + b_0 + c_0, a_1 + b_1 + c_1, \\
                    a_2 + b_2 + c_2)
        \end{multlined}
\end{align*}
    \end{document}

They gives:

yet another possibility -- use aligned within multline:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{multline*}
 [((a_0, a_1, a_2)+(b_0, b_1, b_2)) + (c_0, c_1, c_2)] \\
 \begin{aligned}
   &= [((a_0 + b_0, a_1 + b_1, a_2 + b_2)) + (c_0, c_1, c_2)] \\
   &= (a_0 + b_0 + c_0, a_1 + b_1 + c_1, a_2 + b_2 + c_2)
 \end{aligned} \\
\end{multline*}
\end{document}