Kart is super easy, you basicaly never break. Just let the trottle off and turn very gently, try to be very smooth. When and If you need to break, be sure to be in straight line, same think with trottle. It actually drives very differently from cars, I dont really like it. Maybe it's fun with a wheel.
Speaking of Karts, I can't do them.
I did the beginner ones easily enough, then struggle massively. My cousin came over one day, and wanted a go, so I let him do the next 3 karting races, but he struggle on the final one.
I keep trying, but I always spin out, no matter what I try. I wish SRF was an option in that, just so I could get it done, and forget about it.
I'm not good in many areas of the game, but for some reason, my largest ineptitude is in a light, low HP, automatic vehicle, with only an accelorator and a brake.
I expect most of us that don't drive the Karts well are using the DS3, x throttle, square brake, d-pad steering. It's difficult to feather that throttle and not turn too hard. But no...I don't want to change my controls!
Karts require a delicate touch. Really light steering (usually a jab will do...rarely will you need to go full lock with kart steering). Feather the throttle mid-corner, countersteer briefly if you get sideways (again using slight steering motions).
Somebody up above said you never have to brake. I haven't found that to be true. Braking is needed, but try to brake mostly in a straight line while doing so. Brake, then turn-in.
Victory !!!!! Got gold in the last &%#$@^ advanced kart racing after 10 attempts last night. The secret is to get in 5th or 6th place after the 2nd curve and draft, draft, draft after those little kids. If you don't draft behind a kart you cannot get to max speed (84mph) and they will escape from you. Restart if the kart in front of you spins out, because without drafting another kart you'll never win. Racers will move in a slower pace after you take the lead.
Many posts here mention you don't need to break. I did apply break on straights before a sharp corner when I was faster than 75mph, but drifted a lot when I felt it was ok.