There are certain techniques you can use to gain an advantage over your opponents. Some people say these techniques are cheap - and maybe they are - but sometimes you need a little extra help.
Better Weapons
You may have already noticed that you never seem to get good weapons or items when you are leading the race - it's all banana skins and green shells. Well one trick you try, if you are only leading by a small distance, is to brake just before a set of item boxes so you are in second place when you hit the boxes. This won't guarantee you a great weapon but it certainly gives you a better chance of find some mushrooms or a red shell.
Another technique that takes advantage of this is to stay just behind the leader going into the final stages of a race, giving you the chance to grab a good weapon, such as a red or blue shell, and then use it on the leader just before the finish line so you can cruise through to victory. Yes, it's a bit underhand and sneaky, but sometimes it got to be done.
One problem with this strategy is that sometimes other racers will have the same idea and there will be a little standoff every time you reach the item boxes, as nobody wants to go first and get the rubbish weapon.
Saving Weapons
This is when you hang back in second place, fishing for the better weapons (blue shell, lightning, etc) and when you get one - just stay in second place, close to the leader. Then in the final stages of the race, use the weapon and fly past into first when it's too late for anyone to catch you. This technique only really works when there are just 2 racers left - otherwise you risk losing your weapon to the other racers behind you.