Learn Hardcore Russian: Я купил чеснок на рынке, чтобы улучшить вкус супа. - I bought garlic at the market to improve the soup’s taste.

Купил is perfective: it focuses on a completed, one-time result (I bought it—the purchase happened and is done).
Покупал is imperfective: it’s used for ongoing/repeated actions, background description, or when the result isn’t important (I was buying / I used to buy / I went shopping for).
In this sentence, the speaker means the garlic was successfully purchased, so купил fits best.

Why does чеснок stay as чеснок (no different ending)?

Because чеснок is masculine inanimate, and in the accusative case (direct object) it usually looks the same as the nominative:

я купил (что?) чеснок
So you don’t see a different ending here.

Is it more natural to say купил чеснок or купил немного чеснока?

Both are correct, but they feel a bit different:

купил чеснок = bought garlic (could imply “some garlic” in general, or garlic as an item).
купил немного чеснока = explicitly “bought a little garlic,” using the genitive чеснока to show an amount.
If you want to emphasize quantity, use немного/пару головок чеснока.

Why is it на рынке and not в рынке?

With markets, Russian commonly uses на for places that feel like open areas/events/platforms:

на рынке = at the market / in the marketplace (as a place you go to shop)
В is more typical for enclosed “inside a building” contexts, but в рынке is not normal Russian for “in the market.”

What does чтобы mean here, and why is it used?

Чтобы introduces a purpose clause: in order to / so that.
It answers “for what purpose did I buy garlic?”

Я купил чеснок …, чтобы улучшить вкус супа. = I bought garlic to improve the soup’s taste.

Why is it улучшить (perfective) and not улучшать (imperfective)?

After чтобы, both aspects can appear, but the choice changes the meaning:

чтобы улучшить (perfective) = to improve it (achieve a result; make it better)
чтобы улучшать (imperfective) = to be improving it / to improve it regularly as an ongoing practice
Here the idea is a one-time goal for this soup, so улучшить is the natural choice.

Why is it вкус супа and not вкус супа / вкуса супа—what case is супа?

Супа is genitive singular. Russian uses genitive to express “X of Y”:

вкус (чего?) супа = the taste of the soup
So супа depends on вкус.

Could I also say улучшить вкус супу?

Not with вкус. Улучшить takes a direct object (accusative): you improve something:

улучшить (что?) вкус
The soup is then linked by genitive: вкус супа.
If you want the soup itself as the direct object, you can say:
чтобы улучшить суп = to improve the soup (more general; not specifically taste)

Is the comma before чтобы required?

Yes. A clause introduced by чтобы is a subordinate clause, and Russian normally separates it with a comma:

..., чтобы улучшить ...
You’d only drop the comma in special cases (fixed expressions, very short constructions), but not here.

Does the word order matter? Can I move parts around?

Word order is flexible, but it changes emphasis:

Neutral: Я купил чеснок на рынке, чтобы улучшить вкус супа.
Emphasis on location: На рынке я купил чеснок, чтобы улучшить вкус супа.
Emphasis on purpose (less common but possible): Чтобы улучшить вкус супа, я купил чеснок на рынке. All are grammatical; the original is very natural.

What’s the difference between на рынке and с рынка?

They mean different things:

на рынке = at the market (location)
с рынка = from the market (source/origin; “I brought it from the market”)
Example: Я принёс чеснок с рынка.

Do I need свой anywhere (like “to improve my soup’s taste”)?

Not usually. Russian often omits possessives when it’s obvious from context.
If you want to stress “my (own) soup,” you can say:

...вкус моего супа (neutral)
...вкус своего супа (often used when the subject is the owner; “his/her own” in context)
But the original sentence is complete without it.