In my opinion this is not the best way to go about doing what you really would like to do, which is to get a more serious introduction into science.
(As an aside, the 20 most influential papers will highly depend on the field and interests of the person making the list, so I would say there is no such thing as the top 20 most influential papers.)
Now I don't know much about where you're coming from--what's your background, what are your interests, etc. So I am assuming you are high school or early college age and have a general interest in 'science' but haven't narrowed down to a specific field.
Papers are hard to read even when you are starting off grad school, they almost always require the reader to provide some of the context of the paper and to understand the overall logic that they , and certainly will not explain things from scratch. To really get all the information out of a paper takes a lot of work and thought. In my experience putting a strict deadline on how long it will take you to read the paper will end with you leaving it before you get much out of it. This can leave you discouraged and make it less likely you'll stick to your schedule.
It is also hard to 'learn things' with no specific goal in mind, especially without a specific field in mind. It is a sad fact, but you cannot be an expert in more than one subfield of science, and to really understand the interesting parts of field you have to commit to it and stick with it for a long time.
If you are interested in browsing different fields, I would recommend looking at textbooks for fields you're interested in and getting a sense for what the different fields are 'really' about. What topics do they expect freshmen to know? What problems are people worried about now? What were the major advances in the field? Many of these are probably not popularized heavily so you might not have heard of them even if they were actually extremely important. For example, in theoretical physics, in the 1970s Wilson developed a whole scheme for looking at renormalization that is really the way that modern physicists view this topic. It is enormously important, but few people have heard Wilson's name outside of theoretical physics. You obviously will be able to go into more depth with the intro topics than the research questions, but the idea of this would just be to get a sense of what different fields are about, so you would know 'what you would learn' if you were to pursue that field.
However I would recommend picking the field you are currently most interested in (even if the choice is somewhat arbitrary), and then rephrasing your question as 'what are good introductory materials to learn about X'.
For example, if you are interested in particle physics, and assuming you've had at least high school physics, there are several directions you could go. If you wanted a good grounding in the underlying physics and/or math that you will need to build on to understand more interesting stuff there are great books like Kleppner for mechanics or Purcell for E/M. There are also lecture notes by David Tong that look really good: a survey of modern physics at http://www.damtp.cam.ac.uk/user/tong/concepts.html and an intro to mechanics at http://www.damtp.cam.ac.uk/user/tong/relativity.html. You could look at http://www.staff.science.uu.nl/~hooft101/theorist.html. If you want to dive into more advanced stuff that you can probably tackle early on you can certainly start to learn special relativity. Einstein's special relativity paper is actually pretty readable. The Feynman lectures on physics (which are also great) discuss special relativity as well. There's also a book by French from the 60s which is great, and the previously mentioned Tong notes have an intro to special relativity. In my opinion you also could learn some basic quantum mechanics, but I don't have any resources I can recommend for that. I certainly would not try to read anything about Quantum Field Theory, the Weinberg and 't Hooft paper cited above will be way over your head, which is fine--no matter what field there is a lot to learn about.
The short answer is that there is a ton of stuff to read, but it really really helps to have a specific field in mind. It also helps if you can connect with a teacher of some kind who you can ask questions to... If you're in high school you might be able to find teachers who know their stuff at a more-than-high-school level, but you might not. In college you might be able to find a professor willing to do some kind of independent study. Or you might be able to find people on a forum who are interested in the same things, my impression is that here there are a lot of physics / math / cs people, but fewer chemists and biologists (could be wrong though).