How I read a paper really depends on why I'm reading the paper.
A lot of papers I go to because I have a specific goal. Maybe they have been cited elsewhere as containing a proof of something I want to understand. Or a different proof of something I already know how to prove. Or maybe someone refers to the paper as having a particularly lucid explanation of something. If that's my goal, I go straight to whatever is in the paper that I want, assuming the organization of the paper makes this possible (not always the case, but very common, at least with articles written after say 1950). In doing my close reading of the particular thing I want, I often take notes. Depending on the results of my reading the thing I set out to read, I might go on to read other parts of the paper. But I generally don't.
If I'm reading a paper without a specific goal, I generally start at the beginning and read the introduction. Ideally this contains the statements of the main results, but if it doesn't, I skim the paper for those, and also the statements of any lemmas, corollaries, etc. 19 times out of 20 my reading stops there, because I find that, beyond the statements of the theorems adding to my general awareness of the world, I'm not all that interested in the paper. That 1 time out of 20, I will probably try to come up with the proofs on my own, using the paper's introduction as a really big hint. Whether I succeed or not, I usually end up giving the paper a pretty thorough reading.
So most of the time, for me, "reading" a paper does not mean reading the entire paper, or even reading a large contiguous chunk of the paper. I could probably fit all of the papers I have ever read beginning to end in a regular sized binder with tons of room to spare. I think I am far from alone in this. In fact I think it is probably counterproductive to insist on reading things beginning to end, and I would advise that people who are new to the mathematical literature (e.g. students) suppress the desire to do this. Generally speaking, if you are new to the literature in a given subject, you will waste a lot of time if you read this way.