I would not start with anything coarser than 600 grit, and since plastic is soft, might start even higher to see how well the scratches are removed. You can always go coarser more easily than finer if needed. In other words start with the highest grit that removes the scratches. Remember to use water as a lubricant.
As for my next step, it depends on what you have access to and how much money you want to spend. The cheapest and most convenient source is probably an autobody paint supply store. Ask for "color sanding paper." I prefer 3M but there are other brands. They go up to grits over 6000.
If you have access to3M, you may also consider its "Microfinishing Film." You can get extremely fine finishes with that. Years ago, I used it to flatten (lap) flat and polish glassy carbon to a mirror finish for electrodes. A final polish was not necessary.
Finally, 3M abrasive pads come in various grits. Most people have seen the green and purple maroon pads. The finest pad I have is white. It will polish harder materials but probably leave a little haze on plastic. Since they are pads, you may get a little rounding at the edges and the surface will not be super flat.
Then you need a final polish. There are lots of them available. For very small amounts, consider toothpaste.