I am a teacher and a mom, so I'm always working on trying to stay organized. The latest thing that works for me at home is I bought some "paper boxes" from Ikea that have lids and put them on a shelf in my kitchen with the labels "MAIL" , "COUPONS", and "SCHOOL PAPERS". That way all of the papers that come into the house are tucked away neatly until I have time to deal with them.
For your kids' school work, I would say you certainly cannot keep everything, but also do not want to throw away everything! For special projects that you want to save, you can get one of those art portfolios from a craft store for each child, or I use an under-bed storage box. I highly recommend that you save some of their stories and writing or any research projects they do at school (I know how hard kids work on this stuff and that they are very proud to take it home to share with parents, so it makes me sad to hear that some parents throw EVERYTHING away). Daily homework and worksheets that come home constantly can of course be recycled after looking through it and maybe talking to your child about what they learned, or display one piece from each child on the fridge for the week to show off a job well done. As kids get older, I know some people who have started a binder for each child and they select certain papers, stories, tests, etc. that they want to save in the binder. When they get older, it will be fun to look back at their early work and see how far they've come!
For papers that need to be returned to school, you can either have a bin for "Complete and Return" OR you can keep it in the child's folder (if you're good about going through it every day) and that way it won't be forgotten in the bin at home.
Have a great (and organized) school year!