Greetings: I'm sorry, excuse me, pardon

Anteeksi - I'm sorry
Anteeksi! - Excuse me
Anteeksi? - I peg your pardon

Anteeks - sorry
Sori - sorry
Oho - "oops", sorry man

Greetings: Thank you! You are welcome!

Kiitos! - Thank you!.
Kiitti! - Thanks!
Suuret kiitokset. - Thank you very much! (very formal, see, no exclamation mark in Finnish, we are serious when formal)

Ole hyvä! - You are welcome!
Olkaa hyvät - please (have a seat/dig in/etc.)
Eipä kestä. - Don't mention it.

PS. If someone says "kiitti vitusti", he's being sarcastic and using bad language.

Greetings: Good bye!

Informal:

Moi moi! - Bye bye!
Hei hei! - Bye bye!
Heippa! - Bye!
Näh(d)ään! - See you!

Formal:
Näkemiin - See you again
Hyvästi - Good bye (this one is pretty final and thus rarely used)

Greetings: Good morning / afternoon / evening / night

(Hyvää) huomenta! - Good morning!
Hyvää iltapäivää - Good afternoon. Honestly, I didn't write this with exclamation mark because no-one ever says it enthusiastically. Hyvää iltapäivää is not used often. Instead we just say päivää - good day.
(Hyvää) iltaa! - Good evening!
Hyvää yötä! - Good night!

PS. Kauniita unia, oman kullan kuvia - Sweet dreams, dream about your beloved.

Greetings - Hello!

Finns use quite a variety of "hello"s, depending on area.

Rule of thumb: use the formal one if the situation requires shaking hands. If you can just wave your hand, you can use the informal one.

Informal greetings:
Moi! - very common and easy to remember.
Hei! - Just like "moi", but can also be used to draw someone's attention.
Terve! - Lit. 'healthy', means "I hope you are in good health today".
Moro! / Morjens! - From Swedish "morgen" - 'morning', but is used all around the day. You hear this a lot in Tampere area.

Formal greetings:
Päivää - (Good) day. Used when greeting elderly or just significantly older people, but you can use it to greet your peers as well and it's totally normal.
Hyvää päivää - Good day. Use this when you want to emphasis the 'good' part, or when greeting someone significantly better than you. Like the president of Finland. Sometimes used in sarcastic sense.

PS. In Savo they say "päevee päevee" (päivää) or "hyvvee päevee" (hyvää päivää). I guess that can be either formal or informal.

PPS. There is a Lutheran movement, popular in Pohjois-Pohjanmaa (Northern Osthrobotnia) and Lappi (Lapland), where the members greet by shaking hands and saying "Jumalan terve!" which means 'Greetings from the God!'. People outside the sect they usually greet with "Terve!" with the same meaning.