What statistical methods are best for analyzing long-term rainfall data?
It depends on your data, if the rainfall data is in a grid with temporal and spatial dimension, you can use a NetCDF format to analyse the data via R, matlab, IDL or Python. NetCDF is Network Common Data Form, which is a very common format of grided and temporal data. In Python, you can simply do several 10-year average map to see the long term decadal trend. Or you could plot a transient period of a smaller region with average rainfall to see monthly or seasonally change. To find out the intensity of distribution you could plot a map with a logarithmic scale of colour. Use a threshold to identify extreme rainfall event that is above the threshold. I use a Python package called iris which can easily plot graphs using NetCDF data. You can look at some examples here http://scitools.org.uk/iris/docs/latest/gallery.html

For getting the rainfall data, you can download the rainfall data from the CRU-NCEP database https://crudata.uea.ac.uk/cru/data/hrg/ CRU stand for Climate Research Unit and it has the most accurate historical data available to the public. They are reanalyzed data from meteorological stations and ships. The highest resolution available is 0.5 latitude * 0.5 longitude degree grid. And for temporal resolution they have 6-hourly and daily. The data is in NetCDF format which is a format most map processing data use. If you don't know how to handle ncdf you could use Google Earth as well https://crudata.uea.ac.uk/cru/data/hrg/cru_ts_3.23/ge/ .

To know more how the data is derived you can read this paper. Hope it helps. http://onlinelibrary.wiley.com/doi/10.1002/joc.3711/abstract