Could anyone tell me the most convenient & cheapest way to get from Zhongshan to Xiamen?

There is no direct train that operate between these two places. So you are suggested to take long-distance bus.
Two long-distance buses at ZS General Bus Station can take you to Xiamen South Hubin Bus Station. The schedules are 19:20 and 20:10.
The address of ZS General Bus Station is No.48, Fuhua Road, West Area Avenue. The arrival bus station is located at No.59, South Hubin Road, Siming District.
This way is the chapest and most conveninet way for you to choose.