I need help with choosing a mid-long term forecastic method for this demand I am trying to forecast the demand of a product for the next 36 months, based on its sales history. The demand plot is shown below. 
I honestly don't know what to do with it. I tried linear and non-linear regression analysis but even with the best fitting curve (an exponential curve), the MSE, MAD and MAPE are too high and the residual plots suggest correlation. 
Any guidance would be much appreciated. Thanks in advance!

 A: The first thing to look at is what happened in months 10 and 11: where did the large increases come from? Has the product reached a plateau? If so, essentially forecast a flat line on the level of sales in months 11-17. Or is another increase of a similar magnitude likely? Or was this a one-time push that saturated the market, so that future sales will drop down to the level in months 1-9?
Forecasting 36 months based on 18 months history is very hard.

with the best fitting curve (an exponential curve)

In-sample fit is not a good proxy for out-of-sample predictive accuracy. Fitting an exponential curve to 18 months of history and extrapolating 36 months out invites disaster. You should not be expending more than two parameters on only 18 observations.

the MSE, MAD and MAPE are too high

What is "too high"? Your series may simply be hard to forecast. How to know that your machine learning problem is hopeless?

and the residual plots suggest correlation. 

Given the upward trend, this does not surprise me. The important thing is to understand where that trend came from, and whether the driver is likely to continue.
