You are correct. It doesn't make sense to say "I was like flying", because a person can't normally be like an action. (One could probably come up with metaphorical or poetic examples where someone does this. But I really can't think of one that sounds sensible off the top of my head. You might say something like, "George was like a hurricane that blew through the room" -- but you're not comparing him to the action of blowing, you're comparing him to a hurricane, a thing.)
You could say, "Riding in this tram was like flying." In that case you are comparing one action to another.
You could say, "It was like flying." This is essentially the same as the previous example, with "it" referring to "riding in this tram".
You could say, "I felt like I was flying." Then you're not saying that you, the person, is like flying, but rather that the feeling was like flying.
There are many other possible variations.