In MK8 it was Iggy + Sport Bike + Triforce Tires + Gold Glider iirc
Sadly MK8D changed the stats for every character and vehicle piece, but Iggy and the Sport Bike are still staples for me. Not sure on what tires and glider I want to mainly use with it though.
For 150cc and lower, I'd typically use either Waluigi, Donkey Kong or Roy, with Streetle, Standard Tires and Waddle Wing. This kart combination is what I would consider a jack of all trades, but master of none. At least, it has a nice speed and acceleration stats, so I could quickly recover to a pretty fast speed.
I don't have a kart combination in mind for any 200cc race, but the best advice I could give to you is to avoid having the speed stats to exceed 3 and focus on the acceleration, handling and traction stats. That way, you will have more control to your already super-fast vehicle.
Sometimes I would play with Large Mii or frames that had the same stats like Gold Kart. This gave me the fastest top speed possible and for a 3 acceleration build.
***
On MK8 Deluxe (because tiered acceleration is gone and part stats are different with more variation [YAY!])
I will play with similar but varying builds, often Large Mii with Mega Man amiibo racing suit. No more tiered acceleration is the best thing about Deluxe by far as I can play with lots of different parts successfully. I also found while beating all the staff ghost times for gold unlock, that certain builds favored different tracks and opponents more so than I found the be the case with regular MK8.
This is generally for 150cc. I really was never keen on 200cc, though it is neat they will throw them into online matches if coin total is high enough now. However I don't really change my build for this, and tend to be able to manage 200cc races okay with this large time build.