You could try Mr. Clean's Magic eraser but be warned that it will take paint and logos off. It might even be strong enough to sand off the 16-BIT logo. Be careful using it around the Mega Drive logo. After that, you could try using plain old regular pledge to give it an extra shine to the Mega Drive.
If you live in the US there are some Dollar Tree stores that carry a generic version of magic eraser.
Magic erasers won't work. I've tried them. Maybe if you use it for long enough, but I doubt it. My ones weren't strong enough to take off any logos, though (at least not for the amount of time I used them, but I was at it for ~20 minutes), so maybe there are stronger ones you can buy. The only real way to get scratches out is to use epoxy or epoxy putty to fill them in, sand, then paint. You can also try various polishing methods afterwards (or you could try them separately, but they probably won't do much good). I used a plastic polishing cloth for a while, alternating with a microfibre cloth to wipe away any particles produced whilst I did it, and it improved my Mega Drive a little bit, but I don't think you'll see a drastic difference without drastic measures, unfortunately. This is coming from a guy who was at it for 2-3 hours trying a bunch of different techniques. I'm not saying that there isn't scratch removal stuff out there that will work, but just that you're going to need to be prepared to touch it up again afterwards, and it's probably more trouble than it is worth.