The easiest platinum badge to get is is probably the one for mileage.
You simply need to clock up 200 cumulative miles across all races. By the time you get to the end of the Showcase Tour you'll likely be in the region of 170 miles.
I managed to get this after a six lap GT race at Sebring in the Spotlight Series you unlock after getting done with the Showcase Tour, your mileage may literally vary.
Nothing to stop you retrying a long race over and over, all the miles count.