Mathematica is a software package that does much more than mathematics. Through the 20+ year history of the program it has developed from a text-based mathematics equation-solving and plotting program to a full-fledged graphical user interface front-end resting on top of a kernal that is capable of solving an extremely vast breadth and depth of problems in mathematics, phyiscs, engineering, and most other sciences. In fact, if there is a discipline where data is used in any way, Mathematica can be used to present and/or analyze that data in many different ways. Mathematica can even be used in areas where mathematics is not normally discussed. For example, you can input text from a poem or play and do an analysis on the word count, structure and other possibly interesting things. I've seen some neat examples of this kind of analysis and I will post links to them when I find them.

Here is a list of resources for using Mathematica starting from the basics to the more advanced.

Introduction


Basic advice for people new to Mathematica

Avoid iterative programming using loops like For, use instead functional programming functions Map, MapThread, FoldList ... and pure functions. This makes the code cleaner and faster.


The backbone of Mathematica: Rules (more advanced)


Tutorials


Tips and tricks


Tips for writing fast code


Compile


Advanced evaluation of expressions


FAQ


Wolfram Websites


Mathematica one liner competition


Wolfram Technology Conferences

2011

2010

2009

2007

2006

2005

2004

2003

2001

1999

1997


Blogs


Various Other Websites


Calculus


Forums

MathGroup

StackX

Books


Package for Preparing Publication-Quality Scientific Figures


Interesting (non-free) tools