One of the most daunting challenges to going paperless is figuring out how to get organized. If you don't have a solid system for creating, collecting, grading/commenting, and returning digital documents to your students then you are going to get overwhelmed very quickly. Until recently many teachers used scripts and add-ons to manage this, but it seemed like a lot of work to get started. Back in the spring Google announced a new product called Classroom which promised to make managing documents easier. Rumor has it that Classroom will be available to all teachers on Monday provided their district has it turned on.
From what I have been able to see so far I think that it looks like it is pretty easy to use. I like how it automatically puts assignments into folders then shows who has turned in assignments and who hasn't. You can open the docs and leave comments and you can comment when you return the assignment as well as leave comments that students will see when they open your class. You can post announcements, discussion prompts, news articles, or anything else you want to share.
One suggestion that Stacy made was that elementary teachers might want to set up one class that has all students then have each student set up as their own class to make it easier to assign different assignments to different students.
I will keep tinkering with it so I can help you out if you decide to use it. While there are other services such as Hapara (which is an AMAZING service!) that organize more apps than Google Classroom, this is a good alternative.
If you have suggestions on how you intend to use Classroom or if there are things you have figured out while you have been playing around with it please share them!
Keep in mind that Classroom is brand new! There will probably be lots of changes in it in the coming months and it might not always work as you expect. Be patient as Google works out the kinks.