The most effective way for me is to give talks, go to conferences and workshops, and talk to people. E.g., I gave a talk at Stanford on Friday, and learned much I didn't know about some "aspects" of recent research in number theory from Brian Conrad, while walking back from lunch.
Regarding the internet, there are specialized webpages that sometimes track news for various research communities, as discussed recently herehere. I started reading mathoverflow a few months ago, and currently for me it is bar far the best online way to find out about current events in math research (at least in my area -- number theory). It's just stunning the number of new results and links to key papers I've found on mathoverflow.
Recently one of my students asked me "Why didn't I know that the Sato-Tate conjecture had been proved?" I responded "Do you read mathoverflow?" and he said "no", and it's what I suggested he start doing (instead of reading slashdot and new york time, say). I find out about so much interesting research here. The arxiv is not as useful, since there's no discussions, rankings, etc. Mathoverflow successfully leverages "crowd sourcing" in a way that works for research math, at least better than anything else online I know of.
