Do you really want to build everything from the ground up yourself, or are you thinking of building a pre-fabricated kit car? I think it's great that you want to work with cars hands on and build something, but before you do so, I want you to be at least somewhat aware of just a tiny fraction of the amount of problems that you'll have to solve if you want to completely design and build it yourself.
First off, everything you do, everything you build, must have safety in mind. Cars are dangerous machines, and the wrong thing falling off can mean life or death. With that in mind, are you even familiar with engineering fundamentals? How's your calculus? Have you taken statics, dynamics or mechanic of materials at the very least at the university level? How about fluid dynamics, aerodynamics, thermo, circuits, among a bunch of other university level classes? Do you know how to do engineering analysis like FEA? Are you proficient at using machining tools? Lathes, mills, CNC machinery, welders, tools, drill bits, etc.? Do you have access to all those tools? Do you have the know how to buy the correct raw materials? Do you have a ton of cash laying around or access to sponsorship cash? Some of those things are more important than others, but if you can't answer yes to all those questions, then you either need to find people who do know that stuff to help you, or you plainly just won't be able to build your own race car.
Race cars are typically designed by a team of trained engineers with expensive and specialized equipment and software and a very large budget, for good reason too. Any race car, when broken down is amazingly complex, good cars even more so; the attention to detail and often times, the creativity involved to solve a problem, is pretty incredible.
You don't have your age listed, so I don't know how old you are. But if you're a older person, then I think you should only take this up if you've got spare cash to burn, want to learn something new, and have the patience to actually grind through much of the design details. If you're still young, then I would start by learning about cars and making sure your science and math skills are strong, then go to (or back to) university with a reputable engineering program and get involved in the engineering projects there. If you're really passionate about racing, find a school with a Formula SAE team, like CarerraGT suggested. That will give you great exposure to the process of designing a race car from pretty much the ground up.
Getting the chance to help build a race car is a really great experience, but it's one that I would only do knowing that I'm working with a team of good engineers who have expertise, experience and ingenuity in areas of engineering that I'm not as strong in. If I were you, I would just stick to modding a car to make it race-competitive or maybe building your own pre-fabricated kit car. Good luck.
