Both sentences are correct. Whoever told you that you can't use 'fast' in this way is over-applying a rule. The rule is this: we use adverbs ( like 'quickly') to modify verbs and we use adjectives (like 'quick') to modify nouns. For example: He is speaking quickly. We had a quick conversation. or He drives carefully. He is a careful driver. Because of this, many people wrongly assume that all adverbs end in 'ly'. In fact, there are many exceptions. For example, 'well' and 'hard' are very common adverbs which don't end in 'ly' - 'He sings well' and 'He works hard'. 'Fast' is also an exception because it is both an adjective and an adverb: you can say both 'This is a fast car' and 'This car goes fast'.

Fast is an adjective. Quickly is an adverb. Quickly is correct, but you will use "fast" in much street English.

Fast is an adjective, not an adverb. Adverbs describe verbs, not nouns, so 'quickly' would be the standard answer. However, you may hear native speakers use 'fast' here. It is not wrong, just non-standard usage of the word.

as a native speaker we usually put either "too", "so" or "very" etc. in front of fast in this instance.

Fast is both an adjective and an adverb, in fact, and is not wrong in this usage. https://www.oxfordlearnersdictionaries.com/definition/english/fast_2 https://dictionary.cambridge.org/dictionary/english/fast scroll down on this one to find the adverb entry