Am I reading enough of the scientific literature? Should I read for breadth or depth?
My experience is almost exclusively with mathematics papers, and applies little or not at all to other fields.

Much of eykanal's post applies to math as well, but one big difference is that math papers are much more varied in their structure, not having an actual experiment to tie them together.  A good paper will generally explain its organization in the introduction, however.

One point worth emphasizing is that reading a paper from front to back, trying to understand everything at each step, is usually inefficient.  The most common instance is that a paper often starts with definitions which may be hard to make sense of without understanding the theorems they're used in.  It's generally more effective to skim the paper several times, trying to understand more and more with each pass.

Relatedly, you'll eventually pick up the skill of picking out the most interesting ideas from a paper without reading the whole thing.  Early on, though, it's probably better to read things carefully; it's very easy to fool yourself into thinking you've understood something.

As to your main question, about breadth versus depth, your first priority has to be depth, because that's what you'll ultimately need to be able to do your own research and get your degree.  But if you're learning enough to do that, you want as much breadth as possible.  It actually gets harder and harder to learn completely new things as you get on in your career, even when there may be direct benefits to your research to do doing so.  Laying the foundations of a broad understanding of your field while in graduate school will pay off later.