(disclaimer: not a native speaker, so anyone is free to correct me on any of this)
For the opening greeting, the most typical would probably be as follows (I'll name the recipient Aino as an example):
The first two are, as far as I understand, informal, while the third can be used in either formal(ish) or informal contexts.
If you're writing to a particularly close friend, you can also use "Rakas Aino," which would be the direct translation of "Dear Aino," but I'm fairly sure that Finns reserve this greeting for people they actually consider dear, instead of defaulting to it like we Americans do.
The standard greeting at the end of a letter is:
Note that unlike in English, there is no comma used after the greeting.
Again, if you're writing to a close friend, you can also use "Lämpimästi" ("warmly") or to even closer friends "Ystävyydellä" (not sure if there's an equivalent in English. Literal translation is "with friendship")