짝사랑 is when someone has a serious one-sided liking for another person. If you just start to have an interest in someone named X, you can say: (X-에게) 관심이 있다 or (X-에게) 끌린다 (I am attracted to X). If you're seeing and talking to the person, you might say: (X를 / X하고) 만나고 있다 / 만나다, where 만나다 here means going out with someone. 썸 타다, 썸남, 썸녀 are a recent slang words used jokingly for other people. They might sound awkward if you use it about yourself with a serious face.

Me too, while this is tagged in Korean, the question is in English.......

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Give him a lunch or dinner invitation, as well as have some coffee