- BBC Health
- Telegraph/Guardian Health (and also The Times, but they've firewalled their site so either read it in hard copy or pay for subscription)
- NHS Behind the Headlines -- Really worth it. I would first read a story on the above, and then try to find its parallel here to put everything into context.
- Universities also have 'News' pages where they big up their own research advancements and findings. These are a good summary of what they've done and it's written in an accessible way for the public.
There is NO NEED for you to read any journals such The Lancet, BMJ, Student BMJ, JAMA, NEJM... They are not aimed at the general public and are pitched as scientific documents. Rarely do even pre-clinical medical students keep up with the journals, research methods etc. as half the time it takes ages to decipher all the implications of the statistics into "lay" English.
Reading a bit of Ben Goldacre's column on the Guardian will also do you no harm. Just be careful though when you are reading stories from newspapers, check if they are an article trying to convey (hopefully) accurate information or whether its a columnist putting their own personal skew on a story.