To get the best performance out of your racing bike -- especially if you're actually racing it -- you need to keep the tires properly inflated. This helps make your ride smoother and makes your tires last longer. Check your tire pressure before every ride, even if you're not in the final leg of the Tour de France.
Unscrew the tip of the valve on your bike tire if it's a Presta valve, also called a French or needle valve. Push down on the valve to make sure it's open. If you have a Schrader valve, which is the same as the kind found on cars, no preparation is necessary.
Attach the head of the pump to your valve. If you have a modern pump with two holes, it works for both Presta and Schrader valves. Just use the hole that fits the valve. If you have the kind of pump that requires you to switch the head from one valve to the other, fix it according to your bike's tires when you buy it, and try to stick with those kinds of tires. Alternately, bike and sporting goods shops sell adapters you can use if the bikes in your household use different valves. Read the instruction manual to make sure your pump doesn't require any special considerations. For example, some pump heads require that you flip the lever lock for them to work.
Pump at a slow and steady pace. Don't rush it; you can damage the pump if you're overzealous.
Fill the tire to the manufacturer's recommended tire pressure embossed on your tire sidewall by keeping an eye on the pressure gauge built into the pump. This will often be a range; you can use your personal preference for how your ride feels to determine exactly what works best for you. Your weight also plays a part.
Carry a portable pump with you on your rides in case you need a quick refresher to get home.
Avoid filling your tires from the machine at the gas station. The air comes out too fast and can blow up your tires.