Well, there are two really highly recommended combinations of characters and karts that many players really like to use! They're often considered the best combinations in the game. One combination is with Daisy on her Mach Bike, and the other combination is Funky Kong on his Flame Runner.
The Mach Bike and the Flame Runner are two bikes that are unlocked from the very start of the game, but Daisy and Funky Kong both need to be unlocked before you can use them.
Funky Kong is really easy to unlock. Go to Time Trial mode, post four decently good times on four different tracks (good enough for the message "A fast staff ghost has appeared!" to appear at the bottom of the screen after completing a run), go back to the license screen and back into your license, and then voila, Funky Kong!
Daisy's a bit more time-consuming. To unlock her, go to Grand Prix mode and run well enough on the Mushroom, Flower, Star and Special cups to get at least a one-star grade on each. Here's the grade list:
E (lowest) - D - C - B - A - 1 star - 2 stars - 3 stars (highest)
You need 1, 2 or 3 stars on each to unlock Daisy.
There are a slight difference in these two combinations.
-Funky Kong on the Flame Runner has slightly more speed than Daisy on the Mach Bike, but his turns are much wider.
-Daisy on the Mach Bike has slightly less speed than Funky Kong on the Flame Runner, but her turns are much sharper.
I personally use Daisy on the Mach Bike.
If you have any questions, feel free to send me a PM. :D