When I first got the game I stuck to a kart and used the Classic Dragster. It has good drift and acceleration and helped me get used to the new controls of the game. I wanted something with better control, so I tried out a couple of bikes; the Mach has the sharpest turning.
I used Mario and the Mach Bike for a long time because Mario was my favorite character and the Mach Bike has the best drifting in the game as well as great speed. It is a great kart and I highly recommend it - but it is tough to get used to.
I switched to Daisy on the Mach because she was statistically faster, but I liked Mario's control a little better and switched back.
After growing tired of getting bumped off of the track, I switched to Bowser on the Flame Runner (or Bowser Bike). Bowser has good weight and extra drift statistically and he is pretty much the man. The Flame Runner has a very good top speed, very good weight, very good drift and mini-turbos, and sucky anything else.
As you can see, I experimented a lot. Try out combos you like and see how they work for you.