Can I say 爸, or do I have to say 爸爸?
Sure, there are many different ways to mean "father" in Chinese:

父亲：a formal and polite word, not usual for daily use

令尊：a very polite word to replace "your father" 

家父: equals to “my father” when you are talking with someone else (more polite)

泰山 or 岳丈: a very polite way to call your wife's father，while "岳父" is more common

公公: your husband's father

爸爸 = 爸 = 老爹: just like "dad"