Practice is, without question, the most important part of training. You cannot rely on any upgrades to help you if you do not practice enough.
I did find it a bit strange yesterday when I played for the first time in weeks, attempting training for the first time with my facility and accessories fully upgraded, and I earned two super gold medals and almost doubled my strength combo almost instantly, but I cannot prove that these upgrades caused this effect. I did notice that the Turbo Mode button worked, which did make a difference.
I didn't notice the Turbo button before Arm Candy mentioned it -- it appears when you're actually in a training sequence. For example, once you start "Accuracy" training, the button will appear in the lower right corner. It basically gives you a headstart on your combo count so you don't have to start from zero all the time.
Once you click it, it should toggle from "off" to "on," right? That's all you'll see, really.
But you will notice that you have a head start in a training sequence. For example, if you don't have turbo for the Strength training, you will start at 1 for the first apple, and get one point for each subsequent apple you slice. But if you have turbo on, you get a free 25 points for the first apple.
There might be more advantages, but I'm too busy during a training sequence to notice anything else ....
It should turn from off to on, but if your facility is not fully upgraded, nothing will happen if you attempt to click it.
The most significant advantage to starting training at 25 is not so much the 25 free points, but that you skip the slowest part of the training. If a high combo is reached and then a mistake is made, you do not have to start the training again at a snail's pace. Since the training is never slow, stats are also earned more quickly.
Upgrades don't really help you that much. I've gotten 3/5 super gold medals in a completely new game(okay, I was kind of bored). It's a lot more skill based than not. The strength training absolutely demands the best sword, as the weapon grade determines its range(and range is absolutely necessary).
I never even knew about the turbo button, but it sounds like it would greatly improve practice, as you get to train on the higher difficulties without wasting time.