Well, first off, manual and automatic have nothing to do with shifting, it actually refers to drifting. From what I understand, drifting has always been manual on previous Mario Kart, where you hold a button into a turn to create a drift until the sparks behind your kart change colors and you let go and get a speed burst. So new to Nintendo Wii is the Automatic drift, so Karts & Bikes automaticly drift into corners. From what I have read, Automatic turns and drifts a little better and faster to compensate for the lack of speed boosts, trying to make them somewhat equal.
You drift by holding the "B" or trigger on the back of the remote or wii wheel when going into and holding during a turn. First you will get blue sparks from kart & bikes, when you let go of B out of the turn, you will get a speed burst. For Karts, if you can hold the drift longer, you will get orange sparks and get an even bigger burst of speed. I think karts get that to make up for lack of speed bursts that bikes can get from wheelies.
The purists will tell you that Automatic is for "noobs" and beginners and that its no fun to drive in automatic and some automatic users will tell you there are just as good with Automatic when racing against manual racers.
For the most part, I have done most of my racing on Automatic setting. I have managed to unlock nearly all the staff ghosts and beat a few expert ghosts so far with Automatic (not spent much time doing so) and have an online WFC rating of 6400+ which is not great but respectable.
Its my opinion for all out speed, manual is faster, thus all but one world record comes from manual racres, Moonview Highway being the one exception. So if you are trying to really nail down time trials, manual will be the way to go in the long run.
As far as Grand Prix and WFC racing, I think Automatic can work very well. There are many more variables when playing with other players that may limit the benefits of manual and automatic may have some benefits, so try them out, use what works best for you!!
Below is a video of a guy somebody using their mii character in a wild wing kart on Luigi Circuit demonstrating the drift