Construction Tips

The best source of construction information is the book "Building Robots with Lego Mindstorms" by Mario and Giulio Ferrari.  This book starts with the basics of solid construction, and adds techniques until you are building the most complex robots.  I especially like the hundreds of clear pictures.  The pictures are so clear you can pick up many tricks without reading the text.  If you just flip through the book and look at the picures, the book is woth the price.  The text is equally well written, providing detail without getting bogged down in words.  The book finishes with 20 pages of pointers to additional information, making you an expert on your area of interest.  I strongly recommend this book for everyone.  It is a comprhensive introduction for beginners, and a great reference book for experienced builders.
When attaching a motor to a gear train, use a pulley.  With the smallest wheel
on the motor, and the largest wheel on the gear train, you get very good
speed reduction, and you simplify many alignment problems.

The simplest robot has two wheel mounted in the center.  Use a separate
motor for each wheel.  The robot should be roughly balanced over the wheels.
A slide or castor wheel is placed on each end.

If the forward and reverse directions are reversed from what you want,
simply rotate the plug that connects to the motor.

Reversing polarity on some LEGO lights will change between blinking light,
and light on solid.

Reversing polarity on the LEGO siren will change the sound.

Chain links do not make good tank treads.  They look OK, the they
do not have good traction.

You can make your own LEGO compatible motors.  Start with any small DC motor
in the 6-12 volt range.  Glue flat LEGO pieces on the top and bottom.
Build a jig from other LEGO pieces to get the motor shaft alignment right.

LEGO plates can be glued on many objects, to make them LEGO compatible.
For gluing plates to your own motor, use a thick adhesive like E6000.
To permanently glue LEGO pieces together, use ABS plastic pipe cement,
found in most hardware stores.  Dilute the pipe cement 3:1 with MEK  (Methyl Ethyl Keytone)
If you use your robots much, it is worth investing in a rechargable battery.
You can buy "AA" cells and recharge them individually, but it is a hassle.
Easier to buy a radio controlled car battery, like the one from
Radio Shack, pictured below.  Price is around $20.

 Ni-Cd battery

The easiest car to build has two wheels, located in the middle of the car. 
A separate motor drives each wheel.  Weight is balanced over the wheels.
A skid plate is located at each end, to keep the car level.
If both motors are on, the car goes forward.  If one motor is stopped, and the
other is going, the car turns around the wheel that is stopped.  
The picture below is a bottom view of this type of car.
Home made motors for Mindstorms
A more traditional car can be built using the Lego differential gear set.
This car steers with the front wheels, and has a single motor driving the
rear wheels.  Like a real car, the turning radius is relatively large.
A car with differential rear-wheel drive is shown below.
Differential gear on Lego robot