I'm a grad student in astronomy, and I try to read one paper per day, so I've thought a bit about where to best find good papers to read. Firstly, if you are interested in finding papers in astronomy, the best resource is the ADS search:

http://adsabs.harvard.edu/abstract_service.html

You can search for papers any which way: by author name, by journal, by year, or some combination. Links to the papers themselves are all on ADS. Older papers are available for free on the journals' websites, and more recent papers are available for free on arxiv.org.

When looking for good papers to start with, you can just search for all papers written in a particular range of years and sort the results by the number of times the paper has been cited. (Usually the citation count is a good estimator of the importance of the paper.) If you search for all papers written within a range of years, though, make sure to select the sort by citation count option before doing your search. (Otherwise ADS will only sort the top 200 results that it returns, which will just be the 200 authors whose names come first alphabetically.)
If you're more interested in physics than astronomy, there's also an option to include physics papers in your search.

I would recommend browsing through the most highly cited papers of the past decade or two. When you find one that interests you, ADS will also give you links to all the papers that it cites, and all the papers that have cited it. You can then sort those results by citation count to find another important, related paper. After doing this for a while you start to read through a network of papers in a particular field and get a grasp of how the field has developed and what the current state of the field is.

Another good resource to find papers is review articles. The most important journal for review articles in astronomy is Annual Reviews of Astronomy & Astrophysics (ARA&A). Browse through their recent volumes for an article that interests you, then skim the article. It will give you an idea of the state of the field and will summarize all the recent, important papers in that field. (If it's a good review article, anyway....) Some other review journals that I've used include Reviews of Modern Physics, Living Reviews in Relativity, and Space Science Reviews. (As a note, there are review journals in every field, so this technique works in disciplines other than astronomy. In fact, ARA&A is published by a group which publishes review journals in many disciplines, so you can find many other review journals on their site. I can't speak to the quality of those journals, though.)

An interesting journal that I like to look through every now and again, too, is the American Journal of Physics. It's not meant to publish new physics, per se, but is more oriented towards developing a better understanding of "solved physics." So, for example, in the current issue, there is a paper on explaining how magnetic traps work and another paper which provides a new proof of Bell's inequality. Nothing truly new there, but it can help you gain a deeper understanding of physics, and oftentimes you don't need much background in physics or math to understand the papers.