It'll go pretty fast with an electric sander. Use drywall sanding paper -- it looks like a screen with abrasive on it, so it doesn't load up.

Sand everything down more-or-less flat without going through the drywall paper or tape before applying more mud.

You're also going to need a much wider blade to do the tape. It looks like you have a four or five inch blade. You'll need a ten or twelve inch wide blade. Mix your mud a trifle thinner. It's easier to get right if you put on multiple thin coats, sanding flat in between.

The pros can do it in essentially one go, but that takes an amount of practice I don't have.

A real light orange peel texture will do a lot to conceal imperfections.