If you meet a listener for the first time or you two are not close, you can say to him or her 잘 부탁합니다.
If you need to say that to someone who's close to you, there will be some different nuances, you know.
나한테 잘 좀 해. This sentence has a nuance, like, "You should treat me kindly."
나한테 잘해주면 좋겠어요. "I'd be happy if you treat me kindly"
Actually there is no common Korean expression translated directly as "please treat me kindly." in English, because that sentence would be a bit shaming if someone said that word directly in Korea.