What model can I use to describe the following time-series? I'm wondering if someone might be able to help me locate an appropriate model for the following two time-series (the cyan and blue one, the reds are rolling means).

I'm looking more for a general direction based on the attributes of the data rather than a "use this" answer. Otherwise I'll learn less. Though all assistance is welcome. 
 A: It helps to know what the data is, but you might first consider adjusting them for seasonality effects. Using the rolling mean as a proxy for what the seasonally adjusted series might look like, I would test the two time series for cointegration since it looks like they could share a common trend (alternately, it could just be the result of strong correlation). This would suggest an error correction model or vector autoregression might be a good initial estimation strategy.
A: A possible option that has not been mentioned yet is to try to fit a type of univariate time-series model called an ARIMA model. Without being able to investigate the data further, I cannot say specifically which ARIMA would be appropriate for this data, nevertheless, ARIMA is an option worth considering. The only thing I could say is that it does look as though some degree of differencing would be required to induce stationarity (for both series).
Testing for cointegration, as already mentioned, looks like a good idea. Keep in mind that even if the two series are not cointegrated, you can still use a VAR (rather than a (vector) error correction model (VECM)).
I haven't said which particular model will fit this data (very difficult to answer without having the data!), but I have provided you with three classes of models that are worth investigating. There's actually a lot in that.
A: I would suggest that you peruse this inside-R thread. It has R code, and on the face of it, your problem and that outlined in the link seem similar.
