shine is just a stupid name for the reflector.
shine spiking is when you hit the pseudo-spike someone by using the edge gaurding(hitting someone who is off the edge keeping them from coming back) with the reflector. Notice how it hits downward on many characters, especially Fox and Falco. It hits most characters just enough to keep them from coming back, and can be used repeatedly quite easily by grabbing and dropping down from the ledge.
wavedashing out out of the shine is simple, but sounds complicated when i explain it. Notice how when you use a reflector, you can cancel the lag by jumping out of it. A wavesdash is simply a jump and a quick air dodge into the ground. So you can wavedash out of anything that can be "jump cancelled" The reflector and the shield can both be jump cancelled. (as well as the dash animation if that wasn't obvious already)
DI stands for directional influence. notice how when you are hit by a move, you are hit a certain distance...now if you are holding a direction WHILE (not after) you are hit by an attack, you can influence the knockback of the attack. The most effective DI is a perpendicular DI. If you are being hit by a move that hits you straight up, hold left or right, because this way the amount of distance you travel will be the same, but the line of trajectory is slanted, causing you to go up less than if it were a straight line.(remember the fastest way between points is always a straight line) If you are being hit left or right, hold up so that you will be hit higher rather than forward and off the stage, also giving you more time to recover since you are higher.
