式, 型, and 类 all share similar meanings. You can use them to describe a category, type, classification, kind, model, or pattern of something.
The nuisances can be subtle depending on the context or circumstance it's used. While these characters can be interchangeablely, there are exceptions. As an example let's use a common noun like 人.
人式 can roughly translate to "human/people style/form/pattern", referring to a specific way/system to classify a style of some. Most commonly you'll with practical arts, such as martial arts or cooking (e.g., Hong Kong-style cooking). In Japanese, the equivalent would likely be 様式. If I were a robotic lifeforms and was describing a way of doing thing the human way, could use 式. Both nouns and adjectives can be used here.
人型 can roughly translate to "human/people model/shape." 人型机器人 (android) is a common example of using 人型. 型 refers to the shape and structure of something. 新式 and 新型 can be used interchangeably because they both refer to something new. Regardless of whether or not it's a new style or model of something, the implied meaning is that this is a new type of thing. You typically would only use 型 with nouns as it applies the charateristic of the noun, such as a human-shaped or ball-shaped robot. Only nouns are applicable with this character.
类 is similar to the aforementioned characters as it can be used to describe a some kind of thing. 人类 commonly is used to refer to mankind, but it literally means "human-kind/kin." 新类 is usually used to refer to a new type/kind of type of something (e.g., 虫类). Like a new type/kind of robot (新类型机器人) or new living species (新类生物) of some sort. Both nouns an adjectives can be used with this character.