I am only beginning to learn the advanced techniques described on this board, but here is what I do for training:
Training Mode is good for first learning how to wavedash, L-cancel, etc. Start on 1/4 speed and work your way up. However, you don't have the C-stick in Training Mode, so it will only take you so far. For the new techniques, I've found it helps to start with Mario or Luigi and then go on to other characters like Fox.
Once you get the basics down for the new maneuvers, it helps to do VS matches against Lvl 1 CPUs. Just do 1-on-1, so you have plenty of opportunities to back off a little if needed. Once you're more confident, use a Lvl 5 CPU for an opponent that actually fights back; then you will begin to learn when things like wavedashing are actually strategically useful. Lvl 9 CPUs are useful for improving your reflexes and speed in order to avoid punishment for mistiming moves.
In all VS matches against CPUs, you should consciously avoid taking advantage of the obvious flaws in the AI. The AI always approaches you the same way (i.e., Bowser walks up to you and uses flame breath; Samus walks up to you and tries to grapple beam), and the AI always willingly falls into your charging smash attacks if they are tumbling above you. No human player is this stupid. When you play against CPUs, do so knowing that you are practicing timing and reflexes only; not strategy. Stay on the offensive and don't allow the AI any breaks, or you will be training youself badly.
One thing I do to help myself stay on the offensive is by setting a time limit to a stock match. The time limit should be short enough that it is difficult to completely defeat your opponent in the set amount of time. A good starting point is 1 minute per stock (i.e., 4 stock in 4 minutes, etc.) on a Lvl 9. And then set the time lower. Of course, what time limit is difficult depends a little on the speed of your character, but the point is to work on KOing your opponent as quickly as possible.
Practicing against multiple opponents is useful, too, and should really help you build speed. But you should also be wary of building bad habits based on the faulty AI.
Oh yeah, and I forgot to mention: turn items off, if you haven't already. They can be fun, but they don't help you learn.