The grade doesn't matter. Just have to come in first and get a gold trophy for each cup. So once get gold on the first two cups, another opens. First in that and the last one opens. First in that and you get something nice: new character or more karts.
Mission Mode doesn't unlock anything. But for instance, if you get at least a star in every mission and every single cup on every level, online there will be a star next to your name. Same goes for getting two stars and three stars on everything.
I'm working on the one star right now. Just two more cups in mirror mode I gotta change the A to a star...and I barely touched mission mode so gotta get a star in everyone of those.