Translation:I do not watch a lot of movies.
If an adjective is applied to a verb it's called an 'adverb'. And I believe your right...but it's been a long time since I took syntax and I don't remember if adverbs can apply to the object of the verb or must adhere to the verb itself. And amari IS an adverb and inly an adverb in Japanese...
Adjectives describe nouns - delicious cake, red car, slow train. These are all adjectives. They cannot be "applied to a verb". In English adverbs for the most part are made by adding -ly to the end of adjectives - quickly, quietly, loudly, but there are exceptions where adjectives change to a completely different word (good adj. - well advb.), or seem to have no change at all (fast adj. - fast advb.),to name a few. In Japanese there are only two ways to change an adjective into an adverb. If it is an 'i' or true adjective then the final 'i' is removed and - ku is added to the end. If it is a na adjective then no change is made to the adjective but the particle ni follows it to show it is now an adverb. Adverbs describe verbs. That is they describe how actions are performed - she cooked well, he ran fast, the teacher speaks slowly. They do not describe nouns and so cannot describe/modify an object. あまりis an adverb and it is used with a negative verb to mean something that your don't really do (often) - ie. It is describing the verb by expressing the negative frequency with which you perform a certain action.
あまり implies "not a lot", you cannot use it in a positive clause (well, you can under certain circumstances, but for the moment it's easier if you strongly associate it with the negative). You could also translate あまり+Neg. as "hardly" or "rarely" depending on context. The idea is that there is almost none of something. In practice, it's often just a filler word which tones down the negative a little to make it somewhat less categorical. So if you hear 映画(えいが)はあまり見ません, the speaker might not watch movies at all and just doesn't want to sound blunt (maybe because the other person just said they like movies).
たくさん on the other hand tends to be used in positive clauses, or as an attribute to a noun (e.g. たくさんの人 "many people"). It can occasionally be used in the negative, but only rarely. In that case it actually means "not"+"a lot" (whereas with あまり I would understand "hardly any" or even "none at all". I would then expect たくさん to get the は particle put stress on it. For example, I found an example sentence which in simplified form boils down to:
- そんな国は世界にたくさんはない。(There are not many such countries in the world.)
In this case the focus of the sentence is on たくさん to say there are not many of such countries, but implying that there are some. On the other hand, if the sentence used あまり instead:
This sentence can be translated in the same way as the one above but it’s focus is much more on the countries rather than “many”. The feeling is more like “there are hardly any (maybe even no) such countries in the world”.
In any case, for the moment I think you can safely associate たくさん with positive sentences and あまり with negative ones.