"So I wonder if you use [ like + to + verb] form ?" Yes, absolutely. For example: I am bored, I want to do something to break the boredom, like to eat something or like to watch TV.

I like to figure skate. -- I do it. I like figure skating. -- I do it or I watch it. Figure skating is like dancing on ice. -- similarity Ice sports, like skating, are my favorite. -- skating is an example of the catagory. This is commonly done, but in professional or academic writing, "such as" or "including" would be better. "And I was like, 'get away from me! You are like so gross!'" -- American teenage slang that drives teachers like crazy.

to + verb can only be put after like when like is a verb, not a preposition I like to visit Korea I like visiting Korea You look like your sister

You said "Like is a preposition. So [ to + verb] can be put after " like" as a phrase according to the rule of English grammar." Reread your grammar book. A gerund (verb + -ing) follows a preposition.