Think you have your answer.
But would add that it may be useful to log your data. Then consider doing regular (d) and/or seasonal (D) differencing. The resultant series should be much easier to model. I'm not confident d/D is necessary, but some form of transformation likely is necessary. Hard to tell from graph, but it appears that the volatility increases with time/linear trend.
The models suggested (by the other answers) will give you better forecasts and decomposition of the series, but with some transformation of the time series you can often fit a good-enough polynomial.
(1) As mentioned below, it is unclear if the volatility is directly proportional to the level. If so log transformation is helpful. Otherwise perhaps not.
(2) Square root transformations are underused, but also often helpful in these settings.