I have a time series. I plotted it and saw that it is not stationary. Thus, I have calculated the difference. Then I plotted the autocorrelation and partial autocorrelation on the differences, along with the confidence bands. Non of the lags were out of the confidence bands, and the correlations for all lags are very low. I wanted to ask, does this mean that I cannot make good forecasts, or is there something else I could do. What should be the next stage? I would add and say that my series has no trend or seasonality (there is a trend down which at some point becomes a trend up, like a stock graph).
The procedure that you had following is recommended for specify an ARMA model. Your results show that the series in difference is like a white noise process. Actually this mean that no ARMA model can make prediction better than a long run mean.
For example, results like yours was historically used to achieve conclusion of unpredictability for stock returns. However this conclusion is exaggerated because the results above can lead us to exclude linear auto-predictability but not predictability at all.
Firstly is possible to have better result using others predictor, not only lag values; think about Granger causality (predictability).
Secondly you can try to predict your variable with ARMA tool after some transformation of original data.
Otherwise you can to use model that involve non linear relations among predicted variable and predictors; an example is Artificial Neural Networks.
If your objective is prediction your work is not ending, its starting.
This indeed looks like the end of your tether. I would recommend forecasting either the overall mean, or the last observation, or using Single Exponential Smoothing.
(See here for a motivation for short answers. Longer answers are always welcome.)