There are many groups in the United States and around the world at universities, federal laboratories, and other organizations that work on climate modeling. A few of the larger U.S. efforts are listed here:
National Center for Atmospheric Research (NCAR)
NCAR created the Community Earth System Model (CESM), a fully coupled global climate model that provides state-of-the-art computer simulations of the Earth's past, present, and future climate states.
Geophysical Fluid Dynamics Laboratory (GFDL)
The mission of the GFDL is to be a world leader in the production of timely and reliable knowledge and assessments on natural climate variability and anthropogenic changes and in the development of the required earth system models.
GFDL Climate Modeling Page
Goddard Institute for Space Studies (GISS)
The climate modeling program at NASA's GISS focuses on decadal to centennial climate change with an emphasis on investigation of climate sensitivity, including the climate system's response to such forcings as solar variability and anthropogenic and natural emissions of greenhouse gases and aerosols.
Global Climate Modeling at GISS
National Center for Environmental Modeling (NCEP)
The NCEP focuses on seasonal prediction; NCEP's Environmental Modeling Center is responsible for the enhancements, transitions-to-operations, and maintenance of more than 20 numerical prediction systems comprising NCEP's operational production suite.
Goddard Global Modeling and Assimilation Office (GMAO)
The GMAO, a component of NASA's Goddard Space Flight Center, aims to maximize the impact of satellite observations in climate, weather and atmospheric composition prediction using comprehensive global models and data assimilation.
Center for Multiscale Modeling and Prediction (CMMAP)
CMMAP is an NSF-sponsored Science and Technology Center based at Colorado State University whose goal is to promote ultra-high resolution cloud-resolving global climate modeling.
Note that this is not a complete list of all U.S. climate modeling activities, but rather a sample list of some of the larger efforts.