How to travel from Hong Kong, Tsim Shat Sui, to Shenzhen and cheapest transportation mode?

You can not be dropped off to SZ along the way to HK from Guangzhou.

If you start from Guangzhou, it is suggested to go to Shenzhen first and then use the local shuttle buses to enter HK.
After visiting SZ, you can take the local subway lines to Futian or Luohu Border to enter HK. Then use HK buses or mtr to reach the Tsim Sha Tsui area.

For the best area to stay for shopping and nightlife, which city do you mean? If you want to find such a place in HK, the Tsim Sha Tsui area is the best place. If you mean SZ, Dongmen area is the best choice.