Functional style
Avoid iterative programming using loops like For or Do, use instead functional programming functions Map, Scan, MapThread, Fold, FoldList, ... and pure functions. This makes the code cleaner and faster.
Transpose and dimensions
Something not easy to guess alone at the beginning: if you have x={1,2} and y={3,4},
doing Transpose[{x,y}] or {x,y}ESC tr ESC in the front end will produce {{1,3},{2,4}} (format compatible with ListPlot). This animation helps understand why.
You can also use the second argument of Transpose to reorder the indices of a multidimensional list.
Don't forget to regularly control the output of the lists you generate using Dimensions.
Get familiar with shorthand syntax (@, &, ##, /@, /., etc.)
Programming easily
Getting help: Execute ?Map for example for a short description of a function, or press F1 on a function name for more details and examples about it. You can solve many problems by adapting examples to your needs.
Auto-completion: Start typing the name of a function and (in Mathematica 9+) select from the pop-up auto-completion menu, or press Ctrl+k to get a list of functions which names start with what has already been entered. Once the name of the function is written completely press Ctrl+Shift+k (on Mac, Cmd+k) to get a list of its arguments.
Function templates: In Mathematica 9, after typing a function name, press Ctrl+Shift+k (on Mac, Cmd+Shift+k) and click on the desired form from the pop-up menu to insert a template with named placeholders for the arguments.
Other useful shortcuts are described in the post Using the Mathematica front-end efficiently for editing notebooks.
Use palettes in the Palettes menu especially when you're beginning.
In Mathematica 8, use the natural input capability of Wolfram Alpha, for example type "= graph 2 x + 1 between 0 and 3" without the quotes and see the command associated with the result.