Focus on Tires and Wheels
The best way to improve the speed of a bicycle is to reduce friction caused by the tires on the road. If you replace your thick knobby tires with thinner slick tires, you will reduce the friction on the road and the bike will go faster. On a road bike, choose a tire that is between 19mm and 25mm and make sure that you use the tire pressure recommended for that tire size. On a mountain bike, replacing the knobby tires with slicks will increase the bike's speed, but this is recommended only if you are riding a mountain bike on roads--the knobby tread is necessary for traction on trails.
A bike's wheels are the single most important piece of equipment in terms of increasing the speed of the bike. You want wheels that are strong enough for the type of riding that you will be doing, but as light as possible.
The number of spokes also increases drag, and therefore slows down the bike--so pick a wheel with as few spokes as possible. Deep rim wheels will improve speed as well, but are recommended only if your average speed is more than 20mph. Because a bicycle's wheels and tires impact the bike speed more than any other factor, replacing the wheels and/or tires is the most cost effective way to make a bike faster.
Focus on Aerodynamics & Weight
If you want to go faster on the bike, you need to make sure that the bike fits you well. The best way to do this is to take your bike to your favorite bike shop and ask them to do a fitting session for you. They will check to make sure that your seat height and stem height are properly adjusted, and make sure that your frame size is appropriate for your body. BikeSplit.com has a good description of proper bike position if you want to try to correct your bike fit on your own.
The second piece of making your bike more aerodynamic is adding components that make it easier for you to ride in an aerodynamic position. The first item to consider would be aero-bars. These are bars and/or pads that can be clipped on to your handlebars and allow you to lean forward, placing your forearms on the handlebars, and bringing your body lower toward the frame. Although riding in this aero position will not be comfortable for the entire ride, it will help you go faster, especially into the wind.
Another item which can impact the speed of the bicycle is the pedals. There are many different clips in pedal systems--some are designed for comfort, others are designed to reduce the weight of the pedals and allow a quicker release from the pedal. Speedplay makes very aerodynamic and lightweight pedals that are designed for speed; Shimano makes clipless pedals designed to make it easy to walk in the shoes with cleats. You will need to decide which is more important for your type of cycling as both types of pedals will increase your speed.