I've set of data (time and associated speed) values.
Which is the best tool to solve(predict future values based on past) time serious problem using the AutoRegressive Integrated Moving Average(ARIMA) model?
you can perform ARIMA in SPSS also. SPSS is more user friendly and and you can easily impliment your model in SPSS.
To get an idea on the procedure of ARIMA in SPSS, you can refer my artitle " Effect of Foreign Direct Investment(FDI) on the Economic Development of India” Arthshastra Indian Journal of Economics & Research, ISSN: 2278-1811, Vol.1, No. 3, September-October,2012, page No. 15-26.
The Software doesn't matter, every traditional econometrics or statistics-software should do for ARIMA models. The important thing is that you understand what you are doing. You could easily calculate an ARIMA model in R using the auto.arima()-command. This way, based on the AIC, the best model specification would be fit to your data....But like I said, it's pretty dangerous if you don't really understand your statistical model.
Sorry for this late reply. I just now read your question. I can warmly recommend Time Series Analysis and Its Applications from Shumway and Stoffer. It is a great teaching book with many applications in R and many examples, while staying correct enough in theory.