If you are wondering what modifications you should make to your car before your first track event, and what modifications to make subsequently, read on.
For your first track event, you do not need any modifications. However, you need to have maintenance items taken care of such that the car is in good mechanical condition. There is a separate page dedicated to this: First event car preparation.
Most people with no high performance driving experience do not put significant wear on their car in their first track event, because they do not drive hard enough.
The biggest issue you are likely to face in your first event is running out of brake pads and overheating your brake fluid. To be safe, be sure your brake pads have at least 1/2 thickness left. If you are flushing the brake fluid, consider replacing it with high quality sport or race brake fluid - ATE Super Blue / Type 200 is affordable and will be sufficient for most cars.
The second biggest issue you are likely to face is tire wear. This depends entirely on how hard you drive. If you've never been on a track before, you generally should not wear out your tires excessively. If you have a few track days under your belt you may want to consider a high performance alignment. Consult shops or communities familiar with your car for recommendations.
If your car has reasonably good handling you can enjoy it for a season or more with just high performance brake fluid and high performance alignment, running on street tires. If the car feels good, and you are not breaking parts, there is no need to change anything.
When you feel that the car is holding you back (and this is unlikely to honestly happen in the first 5-10 track events you do) you will want to look into upgrading other components.
HPDE is about becoming a more skillful driver, and not simply going faster on the straights. Therefore you will usually not be hearing about power upgrades from instructors or organizers. Power levels enter play only when you start to compete and prepare your car to specific class rules.
Take some time to think about safety equipment. If your car is a street car, it is unfortunately difficult to install a roll bar, race seat and competition harness and retain the car's usability as a street car. Even with a dedicated track car installing a roll bar can be quite a project. However it's a great investment in your safety, and I must say that as a Miata driver I sometimes feel better in a convertible with a fixed roll bar than in a closed roof car without a roll bar.
A 5 or 6 point harness together with a properly fitting and installed competition seat will greatly increase your ability to control the car. In track driving it is going to actually be more comfortable than factory 3 point belt as you will no longer need to pull or push with your arms and legs to hold yourself in position.
Make sure the brakes on your car are up to the task of track driving. While going fast is optional, stopping is not! Lower power cars can do quite well with stock brakes for several events; the more power your car has, the sooner you will need to investigate and procure track-quality brake pads. Hawk, PFC, Raybestos, G-Loc, Pagid and EBC are some of the manufacturers of race brake pads. Talk to other drivers in cars similar to yours about what they are running for brake pads, or ask for a recommendation from your instructor.
Tires often have the biggest impact on the car's handling, and therefore its fun factor. If you start with all season tires you may want to upgrade to high performance summer tires (Bridgestone RE-71r, BFGoodrich Rival, etc.) or hard R compound tires (Nitto NT-01, Maxxis RC-1). I would recommend against soft R compound tires early in one's driving career because these tires are more expensive and a novice driver will not generally be taking advantage of them enough to justify their higher cost.
Suspension upgrades are very common and come in an enormous variety of options. The thing to keep in mind is that the more expensive suspension parts give the most improvement in the car's handling, and vice versa. Sway bars costing several hundred dollars will make the car quicker but nowhere as much as a race shock/spring/coilover system costing at a minimum $2000 (and usually the higher spec shocks and springs will call for specific sway bars to balance them out regardless).
If you are wearing the tires evenly with stock suspension I would suggest going for a stickier tire before changing suspension. The tires are consumables and if you get ones you are unhappy with it's a much easier decision to revise than getting different shocks. If you are not wearing tires evenly, you may consider upgrading suspension first, in particular to lower the car and potentially increase alignment adjustability.
If you are serious about tracking your car, stay away from low cost shocks/springs/coilovers that are commonly sold and try to obtain suspension parts that are developed specifically for road racing. The "street" suspensions are often deficient in various aspects, and while street driving does not demand much from shocks as well as car balance most people find that sub-$1000 coilover systems are inadequate for track duty. Seek out advice of other people at the track familiar with your car and save your money for components designed for track duty.
With suspension, brakes and tires, and hopefully some safety equipment, you will have a very capable track machine good for many seasons of fun.