いきなり More informal than 突然 or 急に. いきなり could be used, say, to describe someone who proposed marriage 3 hours after he met you. Or a dog jumping out in front of your car.
(きゅうに) Suddenly   $$$
(とつぜん) Basically the same as 急に
ふいに Abruptly – only used about mental processes. Abruptly realize something, or (more usually), to suddenly think of someone you haven’t thought of in a while. To abruptly wonder how your parents are doing.
You can use both 突然 and 急に to describe a sudden downpour of rain, an invitation on short notice (‘It’s kind of sudden, but do you want to go to a movie later today?’) or an out-of-the-blue email.