Brakes are the most important thing on a tracktool. I consider even tyres less important – poor tyres will slow you down, but failing brakes can get you killed.
A misconception is that “bigger is better”. Sure, you can fit Porsche brakes to your Clio, but is it necessary? If your current brakes are sufficient to lock up the wheels/engage the ABS on a dry road, then there’s enough stopping power. You then need to make sure that your brakes can cope with the heat of repeated braking.
One relatively inexpensive way to upgrade your brakes might be to use the calipers & discs from a higher-power version of the same or similar car. You can e.g. fit the brakes of the BMW 330i E46 on the E36 model. Since these are regular spares and not performance parts, they are quite affordable.
If you want to read more about brakes, Grassroots Motorsport has a good article on the subject.
Brake fluid
Standard brake fluid may start to boil under the increased temperatures encountered on the track. This leads to air bubbles in the brake lines that can be compressed, meaning that not much will happen when you step on the brake pedal. Two popular types of high-boiling-point brake fluid are Castrol SRX (expensive) and ATE Type 200 (a bit less expensive). Remember to replace your brake fluid regularly, as brake fluid attracts water, which reduces the boiling point.
Brake hoses
Rubber brake hoses expand a little under pressure, resulting in a brake pedal that feels a bit spongy. Steel-braided brake lines are much stiffer, giving a better pedal feel. These aren’t very expensive, so definitely worth a try. There are several brands out there.
Brake discs
This is where it gets interesting. There are plenty of “performance” brake discs out there, often with holes or grooves that are meant to let gases escape in them. But once you read about actual user experiences, there are many that have negative experiences with the longevity of these often quite expensive discs. My recommendation, bases on what I have read, is to stick to the relatively soft but cheap OEM discs when using the original brake calipers.
Brake pads
Brake pads are very important, and probably affect fading characteristics more than the discs. For a dual-use car, you need a compromise between braking distance with cold brakes and being able to cope with track use. You also don’t want excessive squealing of cold brakes.
EBC are a brand that is often featured in YouTube videos, but I have read many a negative review about their brake pads. People are more positive about Endless, Hawk, Pagid Racing, and Ferodo Racing brake pads. A good set of brake pads is not cheap, but can be used for quite long when coupled with soft OEM discs.
Brake ventilation
Finally, you can try to improve brake ventilation by fitting cooling ducts. Run hoses from the front of the car to the wheel wells and attach them so that they blow air onto the discs and pads from the rear. Make sure that they are not rubbing against the tyres. One example of how to do this is shown in below video.
