Although weight reduction does not improve engine power, it is one of the easiest performance enhancements that you can perform on a car. The reason for this lies in physics where a greater force is required to move a larger, heavier object. Making the object lighter will require less force to move it, but if we use the same force while making the object lighter then the force will move the object faster and further. This is why the power-to-weight ratio, which is calculated by dividing the car's power output by its weight, is a significant measure in performance industry.
When we modify the engine to produce more power, we increase the power output side of the equation while the weight remains the same. Weight reduction, on the other hand, reduces the weight side of the equation while the power output remains the same. We could also work on both ends of the equation by modifying the engine and reducing the cars weight.
Any form of weight reduction on your car's interior would require some compromise on your part. If you're building a full on race car, you'd probably want to strip the interior down to the shell, removing everything from the roof linings to the floor, replacing the latter with a honeycomb carbon fiber part. However, if you're building a part-time racer or a street race car, you may want to retain some of the creature comforts, such as the heater and the noise suppressing door and roof linings. If you intend to carry the occasional passenger, you'd want to retain the passenger seat or even the rear seat.
You could also replace the heavier OEM parts with lighter aftermarket alternatives. Each of the stock front seats, for example, can be 60 to 70 lb heavier than an aftermarket bucket seat. The dashboard and steering wheel can also be replaced with lighter parts. If you want to retain your heater, you could replace the radiator with a smaller aluminum radiator. A smaller heating radiator will hold less water.
There are several removable body panels on a car that can be easily removed. These include the hood, the trunk, the fenders and possibly the bumpers if they are made of metal. These panels can either be replaced by panels made of fiberglass or carbon fiber. Carbon fiber alternatives are stronger and lighter than fiberglass units but are also slightly more expensive. You may even find fiberglass or carbon fiber doors! Alternatively, you can lighten some of these panels by removing its strengthening ribs. On the underside of the hood, for example, there are several strengthening ribs that make the hood one of the heaviest. The same goes for the trunk, which is also quite heavy. You can use either a hacksaw blade or a small angle grinder to cut off the unneeded strengthening ribs, or use the angle grinder to shave strengthening ribs that you want to retain. You can also replace the latch on the hood with aluminum hood pins. With some fiberglass and carbon fiber hoods would also require the use of hood pins to fit them.
There are several bits of equipment in the engine bay that can be replaced with lighter alternatives, removed, or drained of fluid. The extent to which you reduce weight in this area will largely be determined by your budget.
To begin with, you could replace the OEM air box and air filter with a cold air intake that will give better air flow and will probably weigh less than the OEM unit. The radiator can also be replaced by a light-weight aluminum unit while the radiator overflow reservoir can be replaced by a smaller unit. The windscreen washer bottle can be disposed of or drained at the very least. In addition, that heavy 6-cell battery can also be replaced by a lightweight racing battery or by a 3-coil Optima Battery with high Cold Cranking Amps (CCA) if a racing battery proves too expensive.
On the engine itself you could replace the stock cast iron exhaust headers with a ceramic-coated aftermarket exhaust header that will improve performance while shedding unnecessary pounds, but be careful not to damage your O2 sensor. The stock intake plenum can also be replaced by a lighter aluminum plenum. For some engines an even lighter carbon fiber intake may be available. Then the cast iron flywheel can also be replaced by a lighter aftermarket unit or the stock flywheel can be lightened at an engineering shop, as can the conrods and crankshaft.
There is not much you can do to the drive train other than replacing the transmission and differential with a lighter units if they are available for your car. You can also replace the driveshaft with a light-weight aluminum driveshaft.
Chassis and Suspension
Aftermarket coilovers from Koni or Tien will not only improve handling but can reduce weight by up to 10 lbs per axel. Light weight aftermarket TC rods, tie-rods and control arms are also available in aluminum. Replacing solid brake disks with ventilated disks will improve braking while reducing weight. If you've got brake drums at the rear, those may be replaceable by brake disks that would be lighter and more efficient. Then, of course, after market alloy wheels, and aluminum wheel studs and lugs will reduce your unsprung weight, and less unsprung weight improves handling and road holding!