Please advise a fast and cheaper transport (and estimates) from Guangzhou to Shaoxing, Zhejiang.

Hi, you can go to Guangzhou South Yuexiu Bus Station to take a direct long-distance bus to your destination. It takes about 16 h and costs CNY 350. And one bus is in operation each day, departing at 15:50 from Guangzhou.
Similarly, only one train is in service between the two cities.
I suggest you take a train to Hangzhou first. Then you can take a train at Hangzhou to the destination in about 40-45 min. It is more convenient and cheaper.