As the previous poster stated, Amtrak's Auto Train only travels from Lorton, VA to Sanford, FL and doesn't make any intermediate stops except for a refueling/crew change stop in Florence, SC. No passengers are entrained or detrained at at this stop, so Auto Train is pretty much out of the picture.
Also, neither Myrtle Beach nor Conway, SC are served by Amtrak. The closest city served by Amtrak is Florence, SC, about an hour and a half away. If you wanted to travel to the area by rail, you would have to take the Capitol Limited from Cleveland to Washington, DC, then change trains (about a 6 hour layover in DC), taking the Silver Meteor from DC to Florence, SC. Once in Florence, you'd then have to rent a car to drive to the Myrtle Beach area. All in all, it'd take roughly a day from the time you left Cleveland (2:15 AM) to the time you arrived in Florence (3:20 AM the next morning).
If you can, my recommendation would be to fly - you'd still be renting a car once you'd get to Myrtle Beach, but it'd be faster, less hassle, and you'd be right in Myrtle Beach, not an hour and a half away.
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