Generally, adjectives end with ~히 if words they are based of are "~하다-words". Like 조용하다 -> 조용히 or 엄격하다 -> 엄격히 (silent/-ly and strict/-ly). But it doesn't have to be this way with other words. For example 즐겁다 is changed to 즐겁게 or 즐겨. 빠르다 to 빠르게 and 빨리.
Also note, that ~게 is a sort of "universal" adverb ending. As far as I know, 게 can be attached to any adjective (and even verb, although this is a bit more complicated) in order to create an adverb. So: 즐겁게, 조용하게, 엄격하게, 빠르게, 쉽게, 크게, 있게 and so on, and so on...