I would definitely keep all awards! I would also keep projects - research projects, reports, etc. If you have a scanner, you could scan papers as well and keep them electronically, and then you can get rid of the actual papers... for less important things, but things you're not sure you can really part with permanently. If your children have art projects that are, of course, adorable - but too difficult to store - take a photo and keep an "art book" of photos of art projects throughout their school years.
My sister uses a school picture and glues it to a label, then labels a box "First Grade" and whatever fits in the box is what is kept - along with photos, and activity things for events outside of school as well (vacation keepsakes, ballet programs, etc.). She gets the boxes at the Container Store - they're clear and they can hold about 2 inches of depth, but they are wide and long enough to hold quite a bit (maybe 15" wide, 20-24"long, and 2" deep) She stacks them in her kids' closet like a tower with kindergarten at the bottom and they go up from there. It's amazing - and it is easily accessible when projects and such come home and need to be put away.
It's always tough to figure out what goes and what to keep - weekly assignments go, projects and special stories they write or poems, etc. are ones to keep. Also keep report cards in each year's box as well - a keepsake for each school year.
Good luck - great time to get organized!