I compiled the list below primarily for use with research- and lab-based classes I teach.  Feel free to email me if you have suggestions for useful climate data sources that are not listed here!

Mendeley climate change article library

CLIMATE AND CLIMATE-RELATED DATA SOURCES

General climate data (historical temperature, precipitation, etc.)

Climate-related GIS and Google Earth data

Hydrology / water-related data

Paleoclimate data

Teleconnection data

Emissions / greenhouse gas data

Snow and ice data

Ocean data

Drought data

Multiple types of climate change-related data

Climate change impact data / Miscellaneous

Climate Model Projection Visualization Tools

Climate Model Projection Data

Miscellaneous Gridded Climate Data