Learn Hardcore Chinese: Wǒ bùdàn xiǎng qù chāoshì, érqiě yào mǎshàng mǎi shuǐ. - I not only want to go to the supermarket, but also need to buy water right away.

It’s a correlative connector meaning “not only … but also …”. Ideally the two halves are parallel in form (both verb phrases, both adjectives, etc.), e.g., 我不但想去超市，而且想买水. Your sentence is fine: both halves are verb phrases even though the verbs differ. In formal writing, tighter parallelism is preferred; in speech, small mismatches are common.

Is it okay that the first half uses 想 and the second uses 要? Do they have to match?

Yes, it’s okay. 想 expresses desire/plan; 要 expresses stronger intention, need, or scheduled future. Mixing them can reflect a natural contrast in certainty: “I want to go (想)… and I need/am going to buy (要) …”. If you want parallel verbs, use 想 in both: 我不但想去超市，而且想马上买水.

What exactly does 要 mean here—“want to,” “need to,” or “be going to”?

It’s context-dependent. 要 can mean:

intention/future (“be going to”): 我一会儿要买水,
necessity/obligation (“need to/must”): 我现在要买水,
desire (“want to”), especially with objects: 我要水. With 马上, many listeners will hear urgency: either “I need to” or “I’m about to.”

Where should 马上 go—要马上买水 or 马上要买水? What’s the difference?

Both are grammatical but nuance differs:

要马上买水 emphasizes how the buying should be done (immediately).
马上要买水 emphasizes the timing of the plan (about to happen soon). You can also say 马上去超市 in the first clause if the urgency is about going.

Why is there no measure word with 水? When do I need one?

Unspecified mass nouns don’t need a measure word: 买水 is fine. If you quantify, add a measure word: 一瓶水 (a bottle), 一点儿水 (a little), 几桶水 (several buckets). If you mean “to buy water to drink,” 买水喝 is also natural.

Does the location from the first clause carry over? Should I repeat 在超市 in the second clause?

Yes, it carries over pragmatically. Listeners will assume the buying happens at or via the supermarket unless you say otherwise. You can make it explicit: 而且要马上在超市买水, but it’s not required.

Should I say 去买水 in the second half? Does leaving out 去 imply online buying?

Both are fine:

要马上买水 states the action “buy water” (could be anywhere, even online if context suggests).
要马上去买水 foregrounds the going-to-buy action (implies a trip to get it). Given the first clause is 去超市, the destination is already clear, so repeating 去 is optional.

Why is there a comma before 而且? Is punctuation required?

Chinese commonly uses a comma to separate the two coordinated clauses in 不但…, 而且… when they’re full clauses. In short phrases, you might omit it, but with complete clauses like here, the comma is standard.

Can I replace 而且 with 还 or 也? How do they differ?

Yes:

不但…, 而且… is slightly formal/neutral.
不但…, 还… is very common and colloquial.
不但…, 也… is also common; 也 is a bit milder. All are acceptable here: 我不但想去超市，还/也要马上买水. You’ll also see 而且还 together for extra emphasis in speech.

Could I use 不仅 or 不只 instead of 不但? Any nuance?

Yes. 不仅 is very common and neutral; 不只 is colloquial. Meaning is the same: 我不仅想去超市，而且要马上买水 / 我不只想去超市，还要马上买水.

How do I pronounce 不 in 不但? Is it búdàn or bùdàn?

Pronounce it búdàn. 不 changes from fourth tone to second tone before another fourth tone (tone sandhi). Many texts write the underlying tone (bù) even when the spoken form is bú.

Is 马上 the same as 立刻/立即/赶紧/赶快?

Similar, but with nuance/register:

马上: very common, neutral-colloquial “right away”.
立刻/立即: more formal/literary “immediately”.
赶紧/赶快: “hurry up and …”, adds urgency/command. Any could replace 马上 depending on tone: 立刻买水 (formal), 赶紧买水 (hurry!).

How would I negate this idea using the same connector?

Common patterns:

Both negative: 我不但不想去超市，而且不想马上买水.
Counter-expectation: 我不但不想去超市，反而想待在家里. Note the extra 不 in the first clause.

Is 去超市 different from 到超市?

Slightly:

去超市 focuses on the act of going (toward).
到超市 highlights arrival/endpoint and is often used with completion or result: 到超市以后 / 我到了超市. In your sentence, 去超市 is the natural choice.

Do I need 了 anywhere to show imminence or completion?

Not here. 了 marks completion or a change-of-state. To stress imminence you could say 我马上要买水了 (“I’m about to buy water now”), but your original sentence states intention/plan without needing 了.