The Nissan 350Z is arguably the most popular car to customize in the auto tuning world. The car has mass appeal due to it's superior handling, performance, and sporty look. The customization route will differ based on your end goal. You can customize for speed, drifting, handling, looks, or all of the above.
Engine Modifications - If you want to take your engine's performance to the next level, then you definitely want to consider custom headers, intake, downpipe and exhaust. If you are really hardcore you can add a custom oil pan.
Drivetrain Modifications - Most drivers will leave the drive train alone. That is just fine. But if you are racing your car on the track, then drivetrain modifications may be something you should consider. The modifications you would want to look into are a custom clutch and flywheel followed by a 2 way LSD.
Suspension - Most drivers who modify for performance will make changes to the suspension. Suspension modification can add significant gains to your controlability and overall performance. The changes you want to make are: coilovers and rear control arms,strut bar, antiroll bars and tie rods.
Wheels, Tires And Brakes - This is a must for looks and can add some performance enhancement. Your obvious customizations are rims, tires, and brake lines.
Exterior Modifications - No 350Z customization is complete until you add some exterior flair. How else will anyone know you're not a regular Joe? This area is relies more on personal preference than anything. If you really want a different look, you can add a full body kit (front bumper, side skirts, and rear bumper). If you want to stay true to the 350Z's stock look, you can probably get all of the customization you want from a lip kit (front lip spoiler, side skirts, and rear lip.)
Interior Modifications - Some thing that interior modifications are purely for look; the truth is that the right interior gear can increase safety and performance. The interior modifications I recommend are custom racing seats, a harness, and a custom steering wheel. A bolt in rollcage is also great to have.
Perhaps the greatest thing about customizing a 350Z is the wide range of options that you have. Since this car has become so popular, many companies have thrown their hat in the ring and made products for customization. The above information should be more than enough to get you headed down the right path with your customization.