The problem is that relying just on automatic syncing won't save you if you don't check the calendar regularly. Lots of people turn on the connection between Classroom and Calendar but then forget to look at Calendar, so they end up in the same mess as before. What actually works is having a place where you HAVE to see all your assignments when you open your phone. On top of what everyone else says about notifications, try adding a Google Calendar widget to your home screen (you can do it on both Android and iPhone) so you see the due dates without needing to open the app. That way the assignments are right in your face the moment you unlock your phone.
Another thing a lot of people don't do is check Classroom on Sunday night, even if you already have everything synced. Teachers upload stuff at the last minute or change dates without much warning, and five minutes of weekly checking saves you from surprises. If you're the type who gets distracted easily, grab an actual piece of paper (I know it sounds old-school) with the three or four closest deadlines. Writing by hand works better than any app for getting stuff to stick in your head. And if you still miss a deadline after all this, talk to the teacher before they post the final grade - sometimes they're understanding if they see you're actually trying.