I'm the instructor of record, so I grade everything. Since I only have 20 students (yay for small, liberal arts and sciences institution!), I can usually grade fairly quickly.
For short assignments, I grade them immediately after class during my office hours. That's easy for me, though, because they're completion (as long as they actually followed the directions and I can read it). That's probably not so possible in fields other than composition.
For long assignments, I read through all of them, and then sort them by how they "feel," in terms of strong, middle, and weak, and then start grading from the bottom and work up. It's best to save something good for the end, so you're not completely worn out and grumpy. I do use a rubric, but I also scribble comments on their drafts. I did a seminar project on student feedback, and as someone else said: the more you write, the less they read. It depends, though, what the learning outcome from grading is: are you wanting them to revise, or are you justifying a grade? For my first two papers, I don't let them revise and so my minimal marking is even more minimal than normal, just enough to point out a few good things they did and a few bad things they did. Later papers get more detail, because they can revise them.