Relationship between two continuous variables in time series data
You could look at the cross-correlation between the two series for different lags. In short, you would look at the values from one series and see how they correlate with previous (or lagged) values from the other series. Auto-correlation is a special case of cross-correlation where you correlate a series with itself (or its lagged values).

To capture non-linear relationships you could see if you can use one of the series to predict the other. You could phrase it as a forecasting problem and see if adding the other variable improves forecasting accuracy. 

For more details, check out this chapter.