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

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!