The thing about astronomy is that it changes quickly, but not that quickly. The timescale for major changes in our knowledge of astronomy is perhaps ~10 years or so (that's usually how long it takes a big problem to be solved), so it's not as though it's something you need to keep on top of every day or every week. (Unless you're a professional astronomer and want to be aware of all the new papers coming out.) As a consequence of this, textbooks are a good place to start for background. There are tons, but Carroll & Ostlie is a good general introduction to all aspects of modern astrophysics. Each chapter also has general and technical references so you know where to go to learn more about any subjects that you find interesting.
For discussions of current problems in astronomy, the journal Annual Reviews in Astronomy & Astrophysics is a good place to start. Current articles are paywalled, but most of them appear on arxiv.org, so you can search for them there. They contain summaries of all the recent research on a particular problem. If you're interested in learned more about that particular problem, they're a good place to start because they'll point you to all the important papers on the subject.
Another good resource is ADS. You can search for papers here: http://adsabs.harvard.edu/default_service.html You can search for specific papers, or just search for papers with a particular keyword in the abstract or title. There's a handy feature that lets you sort by citations so that you can easily find the most influential papers.
Finally, you can find the newest papers here: http://arxiv.org/list/astro-ph/new About ~50 new papers appear here every weekday. It's a lot to keep up with even if you're a professional, but it can be useful to skim the titles every now and again and see what looks interesting. The introduction to astronomy papers usually acts as a sort of mini-review article. It'll review the problem that the paper is examining and will reference other papers that have looked at this problem. So from reading introductions you can get some background on the problem and the ways that astronomers have been trying to solve it -- and it's a lot shorter than a full review article!
Some of the problems may seem somewhat opaque if you don't have the necessary physics background, but not all of them are, so don't get discouraged! Or if that inspires you to learn more physics, so much the better!
