Summarize:

You can rent a car from the airport and throughout rental locations in and around the city. You will need a credit card in order to rent a car. Enter I-40 and take it as far as Williams, Arizona. This is approximately 420 driving miles. Drive approximately 60 miles to arrive at the South Rim of the Grand Canyon. The total trip time on this route is about 8 hours.  This is the most direct route to the Grand Canyon from Los Angeles, because other modes of transportation require you to go through Flagstaff, Arizona. You can also follow I-40 to Flagstaff, Arizona, and then take Route 180 north to the Grand Canyon. This will take approximately 1 hour longer to drive.
Rent a car in Los Angeles, if you don't have one. Find Interstate 40 (I-40) from your starting location. Take Arizona Highway 64 (AZ-64) north at Williams, Arizona.