How do I know quantitatively if the correlation of two time series is significant?
I believe you are looking for the normalized cross-correlation coefficient (source), which in your case would be computed as follows. First, you de-mean your time series $x(t)$ and $y(t)$ such that

$f(t) = x(t) - \bar{x}$, and
$g(t) = y(t) - \bar{y}$

You can treat your time series as vectors $\mathbf{f}$ and $\mathbf{g}$. Assuming the vectors are of equal length, the formula for the normalized correlation coefficient is:

$R_{\text norm}  = \frac{\mathbf{f}^T \mathbf{g}}{\| \mathbf{f} \| \| \mathbf{g} \|}$ 

This is bound to have a maximum value of $1$ (meaning the time series are very correlated), as you are essentially multiplying together two unit vectors.

However, note that if you have different length input vectors, then you should not scale the output.