Begin a Standard Race Tournament. The Tournament consists of 5 race tracks. The position that you finish in each race will award you points in the Tournament total. (10 for 1st, 8 for 2nd, etc.) To win the Tournament you do not have to win every race, you just need to have the highest combined point total at the end of all 5. If you get 1st place in three races and second in the other two, you'll win.
As you pass between cones on the race tracks, your boost/weapon meter (right side of the screen) will fill up. There are several weapons and gadgets available for you to use. You activate them by pressing the X button. When you have enough on the meter for more than one item, you can toggle your selection by using LB or RB. The first one is the shield. If you are in first place, use the shield to protect yourself from AI racers shooting at you. The straight rockets and homing rockets are also useful.
The B button activates a boost but depletes the meter. Use the boost on straight sections or before jumps. The A button is your power slide. If you can use this effectively to set yourself up for corners, you may never need to boost or use a weapon.