For a long time I wouldn't look at the proof, just the statement. That's one level of understanding. Then I started reading the proofs right away. Many interesting details don't always appear in the statement. That's another layer.
However, if you read a proof straight through, you get the false impression that each step follows logically from the last so the theorem is obvious. Try to explain the proof to your friends or try to prove it yourself. "Reading with your pencil" you can learn which steps are surprising and which are natural.
