Get your own........
Best way is to have your own transport. Rental car agency are aplenty to be picked up from the airport. The cost normally is included when purchasing a package tour from the mainland. It's more convenient. You can plan your trips around to avoid droves of tourists in big buses.
Yes, it's safe! Korea has got to be one of the safest countries in the world, and on tiny Cheju, a small-town atmosphere prevades. Even if you don't stick out your thumb, locals are apt to stop and ask you if you need a ride if they happen to be going in your direction.
This is particularly useful when you're leaving Halla mountain and you have a half hour walk to the nearest bus stop.