For a newbie in this domain, how do we start doing our own toy forecasts (lets say for Bay area)?
http://mathsci.ucd.ie/~plynch/eniac/CFvN-1950.pdf
It's something you can solve in a few lines of python and run with minimal compute resources. A nice thing to try is to feed in some data from somewhere like below and watch what happens as you integrate it forwards in time.
http://www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalys...
There are some available weather modeling packages, such as WRF[0], as a mesoscale (few km to 1000 kms) model, that one can download and run. You'd obviously need to find a resource for the appropriate input data. I don't know how easy it is to get up and running, though.