The simplest micro controller to use is the Picaxe chip, i would consider using a LCD display as it is easier to write to than a LED display as the picaxe can interface direct to a LCD but you need a driver for the LED displays.
As for the code to do the timing it is rather straight forward as all you need is some form of a start signal, that could be a simple light beam across the track and then just run a counter loop in the picaxe program something like...
Main:
Pause 10
inc counter
if counter = 100 then
inc seconds
counter = 0
endif
if seconds = 60 then
inc minutes
seconds = 0
endif
goto main
>>>>>>>>>>
Then its just a matter of adding in the start and finish triggers and a display routine.
In the above program loop "pause 10" stops the program for 10 milliseconds so you will get a timing period of minutes , seconds, and 1/100 of a second.
Inc means increase by 1 (or add 1 to counter, etc.)
All very easy and im sure once you start you will find many other projects for the picaxe chip.
Pete.