The children learn a new language much easier than adults.
The children learn a new language more easily than adults.
I think 2 is right because easy modifies the verb, so it needs to be written in adverb forms, doesn't it?
You are correct that "easy" is modifying the verb and therefore the adverb form "more easily" is appropriate here.
However, there are other errors in your sentence. When talking in the abstract we use plurals with no articles. You correctly used "adults" but it should be "children" rather than "the children" and "languages" rather than "a language".
Therefore your sentence should read:
Children learn new languages more easily than adults.
The children learn a new language more easily than adults.
In this sentence, "easily" is used as an adverb, as it modifies the verb "learn".
The children learn a new language much easier than adults.
In this sentence, "easier" is used as a comparative adverb. So, as far as "more easily" or "much easier" is concerned, both of the sentences use them correctly.
For more information on comparative adverbs, refer to Comparative and Superlative Adverbs on the Oxford Dictionary website.