Hi guys,
I am wondering how do you keep up with the current research. What kind of journals do you read? What are the best journals to keep up with various areas of science (apart from Nature).
Do you have any other ideas how to keep up?

I am coming from the social sciences, and the way I keep is by following specialist news outlets on social media that report latest developments in the field, and I periodically search through the electronic databases for keywords relating to my research. I also do not focus on a specific journal, and focus more on authors, so at times I try to follow the blog of a certain author and keep up with what they are writing in journals and elsewhere. Not sure if any of these tips allow to your field.

That is a terribly broad question - are you looking to keep up with all scientific disciplines (physical, life, medical...) or something more specific to at least your topic area? One shouldn't be too narrow in one's focus, but there is only so much one can 'keep up with'.

If there are specific journals or topics I wish to keep on top of, I register for alerts for their table of contents, or on PubMed (a medical database) for specific topic alerts. To cast the net a bit wider, I sometimes look at what people in my academic Twitter network are sharing. One could also use Mendeley or ResearchGate I think for similar purpose. buru's suggestion of following authors is a good one, though don't get hung up on ones you know at the expense of ones you don't. Still, looking at a good paper by someone you've heard of is a good way to identify related citations/authors/publications you might want to monitor.

Do be aware that sticking to specific journals will give you a narrow/potentially biased view of any given field, as every journal has it's own agenda, so you might not get a complete over-view if you stick to particular publications.

As a science phd student, I generally don't attempt to keep up at all. There isn't time at this stage of my career. I like to work on an idea, performing the literature search at the start of the job. I then don't tend to look at the literature again until it's time to publish. I then start the whole process over again.