There are many videos and tutorials available, but here are my favorites.
3x3 Directions - YouCanDoTheCube.com
4x4 Directions - Solving your Rubik's Revenge
If you are looking for a good, basic speed cube, we use DaYan ZhanChi or DaYan GuHong stickerless cubes in class. I highly recommend stickerless cubes so that you do not need to worry about replacing the stickers when they start to peel. There are MANY other good cubes out there. I buy most of ours via Amazon.