You have to tailor the kart to suit you. I personally go for a balanced Kart with slightly higher acceleration and handling. Acceleration and Handling are most important in my opinion.
There really isn't a best Kart. But there is one that best suites you. You're very limited because you haven't unlocked all the customizable parts. For me personally I like a lightweight vehicle and a lightweight character and is good at handling because I drift a lot
, Do know that each character does make an impact on your Kart. Just test some characters and Karts at random of choose in combinations and see what you like, hey if you don't do good that's fine as long as you collect coins each race you can get more parts AND more experience in each track. That's what I did and found out my best was Shyguy with Bubblejet and *forgot name* but its the very small wheels.
Yeah. In regards to cups, the type stuff and character you use in the cups does not matter, you can even get a 3 Star rank with something such as Birthday Girl/Roller/Super Glider, and that certainly isn't a speedy Kart combo. It's all about your own skill and luck.
I suggest if you want to be serious on it, to practice in the Time Trials for the tracks you are having trouble with. Or, temporarily stop playing in first person or having the Gyro Sensor turned off. Having Gyro Sensor off means you can still play in first person, but you can only turn with the directional stick, not tilting the 3DS around. If you want to know where you can change the settings for Gyro Sensor, it's in the Mario Kart Channel.
Master the art of drifting. The lighter the class, the touchier and sharper the drifting will be. Also, as long as you have a decent combination where everything is pretty balanced, you'll be fine. But I can assure you that unlocking the 150cc Cups without drifting is next to impossible.