IMO the best method is to do some of everything. Read papers in general subjects, talk to the PI about a specific project (that can help guide your reading), attend weekly seminars and thesis defenses and think about the questions that people ask at the end of each seminar. Ask yourself why they asked those questions, (what thought process led them there). Attend journal club if the PI will let you.
In terms of where to start with reading- I don't think there's anything wrong with starting with Science/Cell/Nature papers only, because their impact is usually more obvious. When you pick a field you can look at journals in that field and ask yourself what other experiments they could have done to make that paper better. Practice looking at the figures and analyzing them as stand-alone, to see if you would have drawn the same conclusions that they did if you had gotten those results. Go full-time if you can during the summers because there isn't much time in the school year where you can do science without worrying about tests and grades and stuff.
