What are GOOD ways to travel from GuangZhuo Airport, by road, to ZhangZhou City with Heavy Luggages?

For your destination, do you refer to the one in Fujian Province?
If so, I think the fast and the conveniet method is to fly to Xiamen airport where you can find the direct buses for your destination.
Or you can take the take the Baiyun Airport shuttle bus line 8A to reach Tianhe Bus Terminal where you can find the buses for ZZ directly and the bus journey takes around 8 hours.