General computing literacy for scientists:

Software Carpentry, "Helping scientists make better software since 1997" offers many free video and printed tutorials, starting from basics like UNIX shell commands, and proceeding deep into scientific software issues.

Scientific computing languages and packages:

British NCAS's list of main atmospheric science analysis packages
IDL-Matlab-Python translation sheet (PDF) FROM Mathesaurus

Free:

Python - with modules targeting it for science, earth science, climate science.
GrADS - the OpenGrADS bundle (free)
NCL - NCAR Command Language
R a statistics oriented package

Commercial:

Matlab (if your PI is a paying customer at RSMAS, or buy a $100 student license)
IDL (CSTARS has 25 floating; several of us have individual licenses); Fanning

Web analysis tools:

Ingrid at Lamont-Doherty
Climate Explorer at KNMI, Netherlands
PSD web pages (composites etc.)