I'm not sure if this would really count as "advice" but I can try to tell you how I do it and perhaps that will normalize your own experience.
Deciding What To Read
There's a LOT out there. In fact, there's so much material in my field that I'm not sure it's even possible to keep up with it all. I have about six or seven journals bookmarked. About once per month I skim through their latest contents and read any papers that look interesting or directly relevant to my work.
I also use Google's scholar alerts. They send me a reading list every few days based on my keywords. Sometimes I think I have to get better with my keywords though because I find only about 5% of what they send is of any interest. You can also set up watches on particular authors. There are a few groups that I follow because we work on similar problems.
And sometimes I'm just plain following my nose. A lot of journals I read have suggestions that pop up when I'm looking at one particular paper. Those are black holes that I can get lost in for hours if I'm not careful.
Organizing What You Want to Read
For me, academic reading is a lot like a kid eating lunch at school. First, you go for the desert, then maybe the sandwich if you're hungry. But you chuck that brown squished up banana. (Okay, maybe not the best analogy...) Anyway, it's rare that I read a journal article from start to finish. Normally it's a skim of the abstract to see what they've done, how they did it and what the results were. If that's interesting, I tend to skip to the graphs - and the results. Maybe I'll check out the methods, if the results were of interest. It's rare that I'll read the introduction unless I'm looking for a review of the field or if they're doing something novel that I can't figure out at first glance (kind of like reading a manual).
I tend to download PDFs if I think they'll be interesting, or if I'll ever want to reference them later and put them in my folder system. I have a dozen or so main areas of interest that I'll separate the papers into. Then when I have (make) time, I'll open the folder and read through what I've found interesting. Lately I've been storing things on the cloud (dropbox). This way I can read them wherever I happen to be.
Best Times to Read
I don't know that there's a "best" time for me. I do sometimes feel a pang of guilt when I'm reading "on the clock" but I really try to beat that back because reading is critically important to what I do. It's important to schedule in time for reading, planning and free thinking. Exactly how much of that you schedule in will depend on your personal circumstances though.
I tend to read more toward the end of the day. People leave work, the office quiets down, and if I don't have any pressing matters (I work in a clinical environment) I'll stay for an extra half hour and catch up on my reading.