"My dress" and "My suit" both work. So would Italians simply distinguish this based on context? ( Is "abito" simply some kind of formal wear, regardless of gender? )
I came across this thread that helps explain. It seems like abito is slightly more formal, and vestito may refer in general to a whole outfit... but both are used with both genders. http://forum.wordreference.com/showthread.php?t=337224
(When I google I tend to see more dresses than suits, but you do see suits in both cases. And you can easily add "da uomo" and see plenty of suits.)
I will tell you the common use.
Abito = Vestito = womens' piece of clothes that cover both the upper and the lower part of the body, ending in a skirt (Dress)
Tailleur = womens' two pieces, jacket and skirt, or jacket and trousers, it could be called "abito" but tailleur is more common
Completo = mens' two pieces, jacket and skirt. You can call it as well Abito and Vestito.
Maybe it would make more sense to do an Italian course in your native language, unless it's very rare. Anyhow, "costume" means fancy dress, i.e. when you wear an unusual outfit, which may be especially funny, old-fashioned (period costume) or scary (e.g. on Halloween), in order to look like a particular character, object or a job-role that is different to your normal job. A suit is something many people wear everyday or for serious occasions and is therefore not a costume. Unless you are "dressing up" in a suit for fun as James Bond, Dr Who, or similar!