Locate your favorite key journals. This may vary depending on your own interests and what you have access to. There are many excellent scientific publications that it's worth keeping an eye on. Below are some examples (though this is hardly an exhaustive list!).
Once you have some journals you've decided to keep an eye on, you can set up email alerts or subscribe to the journals' RSS feeds so that you are notified when newly published research comes out.
Press Release Aggregators might also be useful. These contain summaries of research written in language for general audiences. Keep in mind, though, that news aggregators are written by PR offices, so while they're good starting points, if you find something of interest on an aggregator, you will want to look up the original research for yourself. Sample aggregators are linked below.
When you read a new scientific research article or a piece of news about a "new" scientific study, keep in mind that what you are reading is part of a long, ongoing conversation.
"For every paper, there is a rich dialogue going on behind the scenes, involving researchers at many institutions. Within that conversation there are peer rivalries and conflicting points of view. For a paper to get published, it probably went through a level of review and editing by other researchers in the field, who may have raised objections and concerns about things like how the data was collected and/or presented. Just because a paper is published, that doesn't mean its conclusion won't be overturned at some point in the future, by new research. "
--- Mike Orcutt, Research Editor, MIT Technology Review