Question about a timing system.

interrupts are easy too.
Here's some of the basics of what you'll need.
You'll have to clean it up some to get it to compile, is mostly written as pseudo-code to get the ideas across.

How are you detecting the beams being broken?
You could start with a simple mechanical switch, the car hits it & shorts an input pin to gnd to cause the end of race interrupt.

// anything after double slash is a comment

#include <avr/interrupt.h> // interrupts library

// declare your interrupt pins. 2 & 3 are the external interrupts
int pin2 = 2; // int means data type integer
int pin3 = 3;

int car1done = 0;
int car2done = 0;

// for testing to start with, just turn a couple of LEDs on to check the interrupts are working
int LED1 = 4; // pin 4 to 330 ohm resistor to Anode of LED to gnd - write high to turn on
int LED2=5;

// define what you need for the LCD control

// Write a couple of ISRs to do this kind of functionality

//***************************************************
// * Name: pin2Interrupt, "ISR" to run when car 1 finishes
void pin2Interrupt()
{
// disable car 2 interrupts
// if (car 2 not done (== 0) ) {turn on car 1 done light and set a flag that car 1 is done}
// enable car 2 interrupts
// reset the done flags
}

//***************************************************
// * Name: pin3Interrupt, "ISR" to run when car 2 finishes
void pin3Interrupt()
{
// disable car 1 interrupts
// if (car 1 not done) {turn on car 2 done light and set a flag that car 2 is done}
// delay a second or two, then turn off done light
// enable car 1 interrupts
// reset the done flags
}

void setup()
// declare the pins as inputs with internal pullup resistors
pinMode (pin2, INPUT)
digitalWrite (pin2, HIGH)
pinMode (pin3, INPUT)
digitalWrite (pin3, HIGH)

// declare
pinMode (LED4, OUTPUT)
digitalWrite (LED4,LOW)
pinMode (LED5, OUTPUT)
digitalWrite (LED5, LOW)

void loop()
{
// now wait for the cars to start
// will need some attachInterrupt statements

/* Setup pin2 as an interrupt and attach handler. */
attachInterrupt(0, pin2Interrupt, LOW);

/* Setup pin3 as an interrupt and attach handler. */
attachInterrupt(1, pin3Interrupt, LOW);

}