Gold Cup is a lot harder and it will require some weeks of practice.
I recommend RC Bandit for the Bronze tracks, Aquasonic for the Silver Cup, and Bertha Ballistics for the Gold Cup. Every championshio is related to certain ratings. For the Gold Cup you have to race against Advanced cars & Semi-Pros, so it is more convenient to pick a car of the same level. You can unlock some of the missing cars by collecting some stars that are hidden in the tracks in Practice mode.
Also you have to finish 1st place in all the tracks several times in Single Race before trying the championship mode.
If you're interested in improving your skills, check theses tips I wrote to make the pain less stressful: http://www.gog.com/forum/revolt/time_trial_advice
I read your tips and they seem helpful, indeed! After reading your soft turning advice, suddenly ThugsRook control scheme recommendation makes true sense, as accelerating and steering by quickly tapping with the same hand is virtually impossible, unless you stop accelerating altogether and just focus on clicking the steering key you need, which probably doesn't work. Unfortunately, I tried that control layout long enough to realize my brain simply doesn't adapt to having the forward key separate from the other movement keys, as I would often find myself accelerating with a key that did nothing, or trying to steer with a key that either flipped my car or fired the weapon...
I'm also not too keen on the whole "you need to finish every race first place several times before you move into the championships" advice you gave, as that strikes me as boring... championships offer variety, at least, as you move from track to track as the cup progresses. Doing the same track over and over just sounds like a chore, to me, I'm not overly fond of grinding in video games, be it racing games or any other genre. I was hoping I could just play the game at my own pace, but apparently it would seem I'm looking for a more laid-back, casual experience out of a game that's meant to be quite hardcore. Oh, well, I'll be sure to give your tips a chance, even though I'll keep my forward+reverse+steering set to IJKL.