When following another vehicle how much distance should you keep
between you and the vehicle?

When I was a girl learning to drive in New Mexico back in the '50s the
rule was 1 car length for every 10 mph.  Today you would be lucky to
get 1 car length for 70 mph...

I was always taught to have about 3 car lenghts between me and the car
I am following.

The general guidance now is by time rather than distance.  For "rule"
see your region/state's driver's license manual or check with the Dept
of Transportation.

For the guidance, you can see

http://www.smartmotorist.com/tai/tai.htm

http://www.city.toronto.on.ca/taxitraining/following.htm

http://www.kitsaptransit.org/safetyproject/safe_driving_tips.htm

(a selection from serching google with < safe following distance > ).

Be sure to at least double the distance in inclement weather.

General rule of thumb is one car length per 10 mph OR (approximatly
the same distance) two seconds.  Two second rule - start counting when
the vehicle in front of you passes a fixed point such as a telphone
pole and count "one onethousand, two onethousand."  If you pass the
same fixed point before the end of "two onethousand" you are following
too closely.  This works at all speeds.

On the Moterway use the 2 Second Rule:-  

The 2 second rule for a safe following distance: Pick a landmark like
a pole ahead of you, note when the vehicle in front of you passes it,
then count 2 seconds (e.g. "one thousand and one, two thousand and
one"). If you pass the same landmark in less than 2 seconds - back-off
a little, and try again. In wet weather double your following distance
to 4 seconds.

It's from this website:-

http://www.roadsafeauckland.org.nz/PriorityAreas/index.cfm?id=1407