What's the best way to make a car faster?

A car is really only as fast as its driver and which is why the best way to get more handling, control and eventually faster lap times out of your car is too improve yourself! Realistically it is very hard to safely push the limits of a car on the street, so ideally track days, driving school, hillclimbs etc are the places to improve your skills. The knowledge and experience you get from learning the key points when to brake and how to tackle corners will get you going faster than any proclaimed 15bhp mod.

However, if you are confident enough you got the skills of taking your existing setup close to the limit and want to get more out of your car, I would recommend looking into the below 5 basics areas before considering the big and expensive engine upgrades.

Braking

Braking is the most important performance part and overlooked because it's actually the opposite of going fast. However the ability to stop faster in shorter distances means you can go into a corner at higher speeds. The components related to braking quicker and hence improve braking performance are:

Steel Braided Brake Lines
Upgraded Brake Pads
Aftermarket Disc Brakes
At least 4 Piston Calipers, more weight and more bhp might require you to get a 6 piston setup.

Tires

You can have the most perfectly engineered engine, the best driver, and most balanced chassis but if the tires are shit, the whole setup will suffer and go over the place. Obtaining the maximum performance is all tied down to the 4 tires that is the connection between car and road. The ability to accelerate faster, go quicker in corners and stop faster can all be accomplished by fitting high-end tires that are appropriate for the weather, temperature and road surface of the environment you drive in. Some point I would like to make:

Stay far, far away from all-season tires and look for a summer or more accurately a three-season tire.
Tyre dimensions - the wrong tire dimension will affect your speedo and other sensors. If going for a wider tire, make sure the aspect ratio is adjusted accordingly.
AWD Cars - Play it safe and stay with the manufacturer recommended tire sizes front and rear. Having a different rolling diameter front to rear might damage the all-wheel-drive system.

Weight Loss

Less weight will make the car feel agile, react quicker and go faster. The cost to accomplish this can go from free to a set of insanely expensive carbon wheels. Here are some ideas that you can apply to your car:

Replace heavy components such as hood, trunk, and doors to carbon fiber
Lightweight wheels
Lightweight front seats
Remove rear seats
Remove air condition components
Sunroof delete, replace glass and components with carbon or perspex cover.
Replace your glass headlights with plastic versions.
[Drive naked] ... just checking if you read till here :)

Suspension

Lower center of gravity = better aerodynamics / Stiffer car = less body roll

Low center of gravity means weight shifts between tires is less - aka Body Roll. This is important because a tire doesn't give you indefinitely more grip as you put more weight on it, the rubber in the tire is only so strong. This is why tall vehicles such as jeeps have very poor cornering compared to a sports car. Components to look out for:

Lowering Springs or a full set of coilovers
Strut Bars Front and rear, upper and lower.

Air Filter and exhaust

And last but not least something to actually do with the engine. Technically speaking a cone filter and upgraded backbox will only make your sound different. However the car is not any faster, as a wide open throttle is exactly the same in standard and sports modes, it just feels that way because the car might be more sensitive to the throttle. To see a real difference you would need to update the fuel mapping on the ECU and adjust it depending on the climate, installed components and driving style. To make things clear a tuned ECU map for a Nissan 350z in Nevada USA cannot be simply copied to an exact car in Sweden and expected to work straight off the bat, there is different altitudes and temperature to take into the equation.

In summary a $15 ebay cone filter will not increase your engine power to 15BHP without involving an upgraded ECU, Dyno garage and a ECU tune specialist.

Big Leagues - Engine Mods

Once you have all the above catered to you might want to step it up and go pro. For this I suggest you save up some money and then some more and have a look into: