“going with the flow” is literally 흐름을 따르고/타고 있다 (흐름 = flow, from 흐르다 = to flow; -고 있다 = be doing something). There may not be one fixed phrase. These are some examples. - 그냥 (큰) 흐름을 따라가고 있어. - 그냥 대세를 따르고 있어. (대세 = the dominant force/trend) - 그냥 분위기에 휩쓸리고 있어 = letting oneself carried by the ambient mood/force.

You would normally say "just go with the flow" to someone else. You may say "I'm just going with the flow." Example: Tom: What should we do on the trip? David: Just go with the flow! We'll figure it out when we get there. David: I don't care what we do exactly, I'm just going with the flow. Tom: That's the spirit!