What is the best way to say happy birthday and have a nice day to a friend,
I am thinking so far this? is that the best way? Hyvää syntymäpäivää :p hyvää päivänjatkoa, vai yön

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Yep, "hyvää syntymäpäivää" or "hyvää synttäriä" (sounds a bit more childish) is happy birthday. "Hyvää päivänjatkoa" for have a nice day.

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You could also wish: "Mukavaa päivää!" for "Have a nice day!"

And for a birthday you can also say: "Paljon onnea!" or "Onneksi olkoon!" These are good when you actually meet a person, shake hands and congratulate them. But these wishes go for other big days as well: weddings, a newborn baby, anniversary, a new job...

Good luck with your Finnish studies!