Often, papers / reports / ... either give the location of the dataset, or have a link to them. As to the used algorithm, I don't know. The method used is often described, but the actual software and so on is often not given, as that is copyrighted (or whatever the correct term in this case is).
Some random examples of climate studies with datasets are
this one (link at top right),
this one (with also a link to the methodology used), or
this one, or
this compilation of older studies (dataset given as url).
Here the software (code) and datasets are both given.
Another possibility are some datastores, where they have loads of datasets, which in turn get referenced in the papers and studies.
This has a link to one of them, from NOAA. Another one is at the
World Data Center for Paleoclimatology. A third one is the
International Tree Ring Data Bank.