Can colours in Russian take a noun form, or must it always be noun + цвет?
Yes, usually colour adjectives can be used as nouns. Though it might get ambiguous, and then explicit specification "цвет" better be added to resolve the ambiguity. Because, "цвет" is not the only notion those adjectives may represent when used as nouns.
would it be okay to use each in singular with only цвета in plural
Both phrases are correct, but there is nuancing.
Does having the colour names in plural give a different connotation, like that you enjoy a variety of different shades within that category?
In these your examples you better keep the "цвет(а)" specifications, especially if you are like in some shop offering you many different goods. Because without the clarifying, explicitly spelled noun, it might be perceived that you refer to some goods in that shop, that colours are adjectives hinting at the shop offers. Like dresses, or shoes or window blinds or whatever coloured there may be.
Unless you are in the middle of conversation and you were discussing exactly them colours. Then adjectives would inherits that context.
- А какие цвета тебе нравятся ?
- Мне нравятся красный, коричневый и желтый.
In English that reference would probably be made explicit. Compare:
- Which colours do you like more?
- I like red, brown and yellow ones.
Now, if you want to start the colours topic, with no prior context, then you better use middle-gender singular forms of the adjectives, implying uncountable vague categories like "это" and "всё":
- Посоветуй, что бы тебе подарить на день рожденья?
- Сам решай. Но вообще, мне нравится красное, коричневое и жёлтое.
Note how "нрaвиться" verb here switched itself to singular form, because uncountable "всё" is implied here, though not spelled out explicitly.
So, as a rule of thumb, you may think that colour adjectives are still adjectives, but very often they imply some "invisible" noun or pronoun following them. Just think which noun or pronoun would be natural to omit in the given context, and which noun or pronoun your listeners would infer from your phrase. If that is obvious and the situation is informal - then the (pro-)noun would naturally be omitted and only the colour adjective(s) be spelled out. If the situation is formal or there is a possible ambiguity to be resolved then you better spell them both.
Old jokes about Civil War: "Красные придут - грабят. Белые придут - грабят" or "Бей белых, пока не покраснеют! Бей красных, пока не побелеют!". In these jokes referred are Red Army and White Guards, however they are referred as "people", "soldiers" or like that. Like "blacks" about negroid people.
So, you see, colour adjectives MAY be used as nouns, but that would not necessarily be colours themselves those noun-adjectives would mean. It could pretty well be any other notions and nouns implied, providing they can be meaningfully distinguished by some associated colours in the context.