I am used to using MAPE, RMSE, MSE, AIC, BIC, AICc to select the best model for a given time series data. what measures of accuracy is precise for selecting the best method (best model for different methods) using the same series?
There is no best model. It is advisable not to base your model choice on any particular criterion but on a set of criteria. The best model may then be evident.
Please let me recommend this detailed guide to accuracy measures and forecast evaluation:
Chapter Forecast Evaluation Techniques for I4.0 Systems
As mentioned in the above work, the choice of the metric depend on the criteria you used to optimise forecasts. If you applied data transformations and then transformed forecasts back to the original scale, the AvgRelMAE may be a good option.
Generally, before applying any metric, it is also useful to explore your data using some visual tools in order to detect potential data problems, see relevant tools and workflows in the above chapter.