the best way to learn is just dive into it. here are few pointers.
-different cars have different powerbands. if you are having trouble finding the best shift spots, look at the rev while driving automatic. this should give you a ball park as to when to maximize the power of the car. also the video examples in the game also give excellent pointers in shifts as well
-you can actually slow the car down via down shifts (you can hurt a car by doing this in a real car, but luckily this is a game). i usually like to downshift, brake, downshift, etc.
-you may have to look at the tach for now to get the shifts down, but afterwards you should learn to listen to engine rev for shifts.
-have fun