Well, firstly, it is suggested that you fly to Ningbo from Guangzhou. The Ningbo Airport has two direct coaches to Zhou everyday respectively at 11:00 and 15:00 with a fare of 50yuan. On the other hand, you could take the airport shuttle bus to reach the Ningbo South Bus Station where the buses depart to Shenjiamen Bus Station of Zhoushan from 07:10 to 16:40 at an interval of about 40 minutes. The bus fare is 41yuan per person.

Here you could search a flight: https://www.travelchinaguide.com/china-flights/

THANKS TOM
NOW IF I HAVE TO RETURN FROM ZHOUSHAN TO SHANTOU WHCIH MODE OF TRAVEL DO I TAKE

Hi, for the return trip, you could back to Ningbo and take flight to Shantou Airport. But, Ningbo Airport only has one flight to Shantou respectively at every Tuesday and Saturday. Or you may go to Hangzhou from Zhoushan, the Hangzhou Xiaoshan Airport has one flight to Shantou everyday.