SymPy doesn't appear in IPython, but IPython is aware of SymPy if it's installed so you can get nice printing of math in the notebook. Also, why link to the Wikipedia page for IPython and not the homepage?
For most people, I'd actually recommend the use of the IPython notebook for doing any SymPy work as you can get the nicely formatted math and inline plots. If one is doing debugging, then you're better off in one of the Python IDEs.
The PyDy guys who work on the sympy.mechanics module have done some nice IPython worksheets.
The SymPy tutorial that's part of the SymPy docs isn't too bad and there's the SymPy live shell so one can try things out.
While we've mostly moved away from IRC, we're now on Gitter and there's also the mailing list if you have specific questions.
Note: I'm a SymPy developer (and a GSoC 2014 mentor).
It's not a program, it's the name of a Python library responsible for mathematical functions:
http://sympy.org/en/index.html
Quote: "SymPy is a Python library for symbolic mathematics."
> Surely one need not explain the etymology of the name of a program in order to give a tutorial on it?
It's useful to know what these terms mean.
> ... but the author's advice appears to be quite specific to SymPy.
Yes, no problem there. SymPy's abilities appear in IPython (http://en.wikipedia.org/wiki/IPython) and Sage (http://www.sagemath.org/) to some extent also -- I've been involved in these projects for years.
My Sage tutorial: http://arachnoid.com/sage
My IPython tutorial: http://www.arachnoid.com/IPython
My Python tutorial: http://www.arachnoid.com/python