I hope by now you have subscribed to a bunch of topics using Google Scholar, including any citations to your own work. This will keep you informed on topics closest to you, even if you don’t read the papers through.
For teaching and future projects, it is a good idea to be more broad. One way of doing that is to subscribe to tables of contents of a few of your favorite journals. It takes only a few minutes to browse though those lists, occasionally clicking on the most interesting looking titles.
If you are responsible reviewer, you will see a lot of new work that way. If you are not reviewing as much as you would like, please let your friends on editorial boards know.
If you are writing a paper or a proposal, odds are you are reading intensely in a few areas.
In my Dropbox so I can get at it anywhere I have a file called Read This!. It is supposed to let me read articles effectively. If a paper is in that box, I have read the abstract. Sometimes I get to the rest of the paper, but mostly they languish there, gradually falling down the list.
But why do I say you don’t read enough if I struggle with it myself? Because it is a constant challenge. Because I don’t hear conversation on papers as often as I would like. Because students discussing their time use do not usually bring up reading papers.
It is all right to read some things more thoroughly than others. Even reading titles can give you an idea of what is out there. Keep reading through the abstract or even turn to the figures and you are likely to get most of what you need. So read thoroughly and carelessly. Don’t necessarily keep track of everything. The mind is a complex place where ideas can sort around and stick together in exciting ways. Maybe if reading were allowed to be haphazard and fun you would do more of it.
Sociobiology 
Here at WUSTL we have subscribed to an app for reviewing current tables of contents on mobile devices, BrowZine. it handles authorization for fulltext access quite gracefully. [A web version is expected sometime this summer.] Many university libraries have this available. So if you don’t want to fill your email inbox with current journal tables of contents you might want to try this free app. More info for WUSTL folks: http://libguides.wustl.edu/browzine
I frequently tell my students that every article they presents a learning opportunity. Some will be exciting and some plain old boring. While it is natural for us to focus on the ‘exciting’ ones, sometimes the boring ones will help them identify their interests or, in the worst case scenario, how ‘not’ to write an article.
I had a question regarding your statement…. “If you are not reviewing as much as you would like, please let your friends on editorial boards know”. But what if I don’t have friends on editorial boards? How do I get on a reviewer list in that case?
That is a good question regarding how to get to review papers. I have two suggestions. First, contact the person that has handled your own published papers and let them know that you are happy to review manuscripts in your field. If you have not published, you are unlikely to be chosen to referee, unless it is by helping your adviser with her reviews. I will write more on this topic in a separate piece.