The first thing you want to do with the professor is establish a reasonable late policy. We emphasize the word reasonable, because this is the policy you will have to live with for the rest of the quarter. It should be reasonable to you so that you can grade assignments together instead of having them spread out in time. It should be reasonable to the students, or at least fair and consistently enforced. Be aware that in lower-level classes, students will try to hand in assignments as late as possible, so don't be too lenient.
Just as you establish a late policy for your students, you should also establish a late policy for yourself. Don't let ungraded homeworks heap up on your desk. Try to the best of your ability to return the homework within a week. Try to figure out how much time you are going to be spending grading and allocate a block of time to do it. It is generally better to do it all at once so that your grading is efficient and consistent.