In many languages, people often say, "Have a nice day" after saying goodbye. This sentence translated in Chinese is, "Zhù nín kāi xīn 祝您开心" or "zhù nín dù ɡuò yú kuài de yì tiān. 祝您度过愉快的一天" But frankly, in Chinese culture we don't say this. So, what do we say instead? It really depends on the situation. Here are some examples:
When you are in a store or in a restaurant, when you leave waiters will say, "Huān yínɡ xià cì ɡuānɡ lín 欢迎下次光临" (welcome to next time!), because they want you come back as frequently as possible.
When you are going to travel, maybe your friends will say "Zhù nǐ/ nín lǚ tú yú kuài 祝你/您旅途愉快" (have a nice trip) or "Zhù nǐ/nín yì fān fēnɡ shùn 祝你/您一帆风顺 " (smooth sailing).
When you are a guest in someone's home, the host usually will say the following as you leave: "Màn zǒu/hǎo zǒu, lù shɑnɡ xiǎo xīn. Yǒu kōnɡ ɡuò lái wán ér 慢走/好走，路上小心。有空过来玩儿 " (take your time/ wish you to be good and careful on the road. Come by again when you're free!) to show that they care for you and to express their enthusiasm.
When children go out to play, perhaps their mother will say, "Wán de kāi xīn, zhù yì ān quán 玩的开心，注意安全" (play good and be careful).
If you want to say "have a nice day" to an elder, you can say, "Nín bǎo zhònɡ shēn tǐ, zhù nín fú shòu ān kānɡ 您保重身体，祝您福寿安康 " (you take care of yourself, I wish you good health and longevity) to wish them a long life.
An elder might express the feeling of"Have a nice day"by saying, "Lù shɑnɡ xiǎo xīn, zhù yì ān quán 路上小心，注意安全" (be careful on the road and be careful [in general]).
"Have a nice day" between friends is, "Yǒu kōnɡ zài yuē, huí jiàn! 有空再约，回见" (Let's meet when you are free. See you next time).
So when in another culture, even wishing someone a nice day might not be understood because it may not be an expression that is used. For more culturally appropriate expressions visit.