It's a good question. The nouns abbigliamento vestito abito indumento behave differently when singular or plural. Don't confuse the noun vestito with the participle from vestire.
You correctly observe that vestito (singular) quite likely is a dress if referred to women, a suit if referred to men. In this meaning, abito is almost a synonym. For instance abito da sera is more frequent than vestito da sera, but both can be used. The sentence
La duchessa indossava un abito nero.
conveys the idea that the duchess was wearing a black (probably one-piece) dress. However, if I want to refer to the famous Audrey Hepburn’s black dress in “Breakfast at Tiffany’s”, I'd use abitino nero.
When plural, vestiti and abiti may refer to several formal clothes for men or women, but they also have a generic meaning:
Sulla spiaggia i ragazzi si tolsero i vestiti e si tuffarono.
In this case, vestiti is just the same as “clothes”. In this sense abiti is still a synonym, but more “written language”. In the singular, but also in the plural, they denote clothing or pieces of clothing that cover the main parts of the body (possibly leaving arms or legs uncovered).
What's indumento? According to the Treccani dictionary it's any piece of clothing. However, nobody would go to “comprare un indumento”, because it's too generic. An example:
Un maglione islandese sarebbe proprio l'indumento adatto per questo freddo.
One cannot use vestito or abito in this sense, because we're referring to a specific piece of clothing that only covers the torso.
The boys above might also have tolto gli indumenti, but I'd not use this form myself.
In the plural indumenti can be a collective noun: indumenti intimi is the underwear, indumenti da lavoro are work clothes.
What about abbigliamento? According to Treccani it is “way of dressing” when abstract, the set of objects and accessories for dressing when concrete. Thus a negozio di abbigliamento is a shop where you go to buy clothes and accessories such as belts. Not shoes, although nowadays you can find shops that sell both clothing and shoes. In informal language one can also say negozio di vestiti, but I think to never have heard an “absolute” negozio di indumenti; negozio di indumenti intimi can instead be heard.
Vestiario is the set of clothes worn or owned by a person, so magazzino di vestiario is, for instance, where a company or organization keeps the clothes to distribute for work (think to the army or a hospital).
While negozio di vestiario is pretty similar to negozio di abbigliamento, the former conveys an idea of lower quality.