"Goes down well" is not a commonly-used expression, and I wouldn't use it for comedy movies.
If you really insisted on knowing the right way to use this phrase to express your fondness for a particular movie genre, you could try something like:
I like many kinds of movies, but comedy movies go down especially well.
(Notice how, because movies is plural, we change the singular goes to the plural go.)
The phrase is a figurative speech for food or drink. If some particular kind of food "goes down well," it means you enjoy it while you eat it, and going down in this context refers to swallowing.
Although I can appreciate you wanting to expand your vocabulary, I'd be very careful with this phrase. For one thing, the phrase "go down" can be used in many different ways, and the phrase "go down on" means to perform oral sex on someone.
I would stick with variants of words such as enjoy:
I find comedy movies very enjoyable.
There are some synonyms on your original link, too. I like "have an impact on":
Comedy movies have an impact on me.
But you would only want to use this phrase if comedy movies were more than just enjoyable, but had some kind of effect on your life somehow.