It is important to understand the grammatical differences between these words.
Clothes: This is a PLURAL word, referring in general to what we wear. It refers to shirts, trousers, dresses, socks, etc collectively. It has no singular form.
Clothing: This is an UNCOUNTABLE word with a similar meaning to 'clothes', but it is less common.
Garment: This is a COUNTABLE word referring to one single item of clothing.
A shirt is a garment. Two shirts are garments.
Costume: This is a COUNTABLE word to describe an entire outfit worn for a specific purpose. Nowadays, we use 'costume' for what people wear when they want to assume a different identity, and it can include wigs and make-up. For example, actors wear costumes on stage.