Experience has helped me get better at quickly finding the most relevant parts of the paper and at understanding what they mean without needing the rest of the paper for context.
I started out in undergrad reading papers straight through start to finish, since that was the obvious default thing to do (plus it felt like what I was supposed to do with a paper that was assigned in class). I gradually developed this approach over time. So it’s somewhat hard to know how this approach would’ve worked when I was first starting out; it does at least seem worth trying (and thus worth sharing here).
Also, my description is just an approximation of what I do, which varies in complicated-to-describe ways from paper to paper (and depending on what I want out of the paper). And much of the work is in picking which papers to look at, and in getting background information from other sources (like Wikipedia).